The morning air sparkles over the sunlit Gulf of Mexico but thickens with grayish smog even on a Sunday morning as the jet swings over the long valley where Mexico City lies.

For what seems an age, one flies over the ocean of pale, concrete urban sprawl that is one of the world's biggest cities.

No new water supplies have been found since 1972. Giant pumps must lift supplies over a 3,000-ft. mountain. In one slum, Netzahuacoyotl, 4 million people breath polluted air on polluted streets. The population of the entire city is 17 million.

Every day some 1,100 rural Mexicans trek into slums such as Netza, as it is called, looking for work.

''If the peasants can read, they become domestics,'' says Gloria Lopez Paz, mother of four, as a daughter hauls rainwater from a grimy well to wash clothes.

''If they can't, they die. There are no jobs. They don't know how to watch out for cars. They don't know where to go for a doctor. They live on the street. They drink. They abandon their children ...'' - in the hundreds of thousands. In Brazil (population 132.6 million) the number of abandoned children is said to be 20 million.

Pollution from 2.8 million automobiles and from industry hangs in the thin, high-altitude air. Lead levels are three to four times above safety levels. ''Don't jog while you're here,'' a friend advises.

Cairo, the biggest city in Africa and in the Arab world, pulsates to the blare of car horns. A taxi changes lanes at a fast clip, ignoring the rear-view mirror. Ahead, a red light, a solid phalanx of cars, and a long wait.

As the minutes tick away, the driver of a nearby car is riveted to his mirror , which reflects the flickering image of a portable television set mounted in the rear window.

Cars in central Cairo park nose-to-tail, forming a virtual fence of aluminum which keeps pedestrians from the road. ''I can hardly believe how much the traffic has worsened,'' says an Egyptian United Nations official just back after eight years abroad.

Designed to hold 2.5 million people, Cairo now is a city of 8.5 million, and another 2 million crowd into the city to work every day.

According to the UN, 10 million people will scramble for a living here by 1990, and 13 million by the year 2000. This would make Cairo the 13th largest city in the world. But Aziz el-Bendari, chairman of the state Family Planning Committee, says Cairo will enter the next century with 16 million people, twice the number it has today. Tens of thousands of squatters already live among the mausoleums in the Necropolis, Cairo's cemetery area.

When Lennie Kangas, a senior US population official, meets Egyptians he tells them he first came to Egypt in 1963 - ''20 million people ago.''

These are just two examples of third-world cities growing faster than their governments and inhabitants can keep pace.

Others include sprawling Bombay (10 million today, 12 million by 1990, and 17 million by 2000, according to UN figures); Jakarta (7.3 million today, 11.4 million by 1990, perhaps 16 million by 2000); and Sao Paulo (almost 19 million now and almost 26 million by 2000).

These and others are the most visible proof of the impact of rapid population growth. Overheated slums throw together the urban poor and newly arrived immigrants from the villages.

Despite a drop in world growth rates in the decade to 1984 (Mexico fell from 3.5 to 2.5 percent a year in the 10 years to 1982), a number of factors have combined to push up absolute numbers of people in the world faster and faster - especially in the cities.

One factor is better health care, which has lowered death rates, while birth rates have stayed high. Another: Almost half the urban growth in developing countries today comes from millions of villagers and farmers giving up bleak rural life and hoping to find work and opportunity in the nearest glittering big city - Mexico, Cairo, Jakarta, Karachi, Sao Paulo, Delhi.

A new era has begun in the third world. Instead of rural areas growing fastest, as they were still doing between 1970 and 1980, cities now lead the way.

So tremendous pressure is building on city officials to provide new services such as housing, water supplies, and schools - at a time when world trade has been falling and third-world debts mounting.

Also being heard are calls for family-planning services to be expanded in urban as well as rural areas, and for efforts to create smaller urban centers to siphon off the flood to the supercities.

In 1800, only 3 percent of the world's population lived in cities. By 1920 the figure had risen to 20 percent. By 1980 it was 41.3 percent. And by the year 2000 one person in every two will live in a city.

Sao Paulo, which could be the second biggest city in the world by the year 2000, was smaller than Manchester, Detroit, and Naples 30 years ago, the World Bank points out. London was the world's second biggest city in 1950; by the end of the century, London won't rank among the top 25.

One out of every four South Koreans lives in Seoul. Baghdad is home to 35 percent of all Iraqis.

The way cities are exploding is best illustrated in Latin America, where seven out of every 10 people are now urban, according to Robert Fox, noted Inter-American Development Bank sociologist and demographer.

The growth looks likely to continue, Mr. Fox says. In Mexico, Central America , and Brazil the countryside offers little hope or money to the peasant, who doesn't own the land he works.

''The middle class moves to the city to look for opportunity but most immigrants are the landless poor, who go to forage and live off their wits,'' says UN demographer Dr. T. Krishnan. ''They don't give to a city. They take.''

What are the major problems to be solved? What is being done about them?

* Housing. In Cairo, rents are as low as five to 10 Egyptian pounds ($7-$13) per month in the older, crumbling apartment houses. Landlords say they can't afford repairs. Janitors and their families live on the roof, under sheets of iron held in place by stones.

In Lagos, dozens of families cram into the same apartment house, sharing a communal bathroom: The city needs at least 2 million new housing units immediately.

At dawn in Bombay, one sees bundles of rags lined up along narrow bridge parapets, high above the water. It is a shock to realize that the bundles are sleeping people. The rags move and people sit up, rub their eyes, slowly gather dried cow dung for fuel, and light fires in air already thick with the smoke from others.

It is a tragic sight. Slum dwellers and squatters account for 46 percent of the people in Mexico City, 79 percent of those in Addis Ababa, 60 percent of Cairo, 67 percent of Calcutta (but only 26 percent of Jakarta.)

City officials try to build more housing. In the legendary beauty of Sri Lanka, Prime Minister Ranasinghe Premadasa is pushing ahead with a plan to build 1 million new dwellings by 1987.

The mayor of West Jakarta, H. Eddy Ruchijat Soheh, says housing and land are his biggest problems. Squatters are hard to evict from vacant land.

His region contains 1.3 million people, he says, and is growing at 10 percent a year. Only 28 percent of homes have piped water.

One solution: to build more high-rise housing - if he can obtain the funds.

In Calcutta, city officials are trying to give as many sidewalk dwellers as possible a single room with hard floor, and access to power and water. They feel it is the best they can do.

* Jobs. A surge of young people is on the move in Latin America looking for work.

In 1950 the number of workers in Latin America was 55 million. By 1975 it was 97 million. By the year 2000, it will be 197 million, says the Inter-American Bank.

This means the need to find an extra 4 million jobs a year. ''And these figures are firm,'' says Robert Fox. ''The people have already been born.''

That is way beyond anything achieved so far. The US itself created 2 million jobs a year through the 1970s. The prospects are for higher unemployment in the hemisphere and more underemployment.

Taken together, the combined labor force in Mexico and Central America will more than double from 22.4 million in 1975 to 52.6 million by the year 2000 - and quadruple 25 years later, to 89.4 million. About 1.2 million jobs a year need to be created in this region alone - but the area's economy is only 8 percent as big as the US.

Elsewhere, in countries such as Algeria, the Dominican Republic, Jordan, Lebanon, Malaysia, Morocco, Nigeria, and Syria, the labor force is expected to double between now and the end of the century.

Jobs depend on new industries as well as on government offices and projects. A number of presidential advisers in Washington say that private industry should be given free rein to create jobs. Third-world officials say it isn't that easy.

''Your Reagan administration tells us to develop our industries and not to expect too much foreign aid,'' Emil Salim, Indonesia's minister for population and environment, says. ''So we use our low-salary workforce to get into textiles - and you put such high import duties on our products that we cannot sell them to you.

''We don't have the roads, power, telephones, or schools to attract enough of your private investment, which prefers Europe.''

Mr. Salim and many others in the third world see one effective answer to city and development problems: proper family planning. Smaller families would reduce the population growth that is diverting money to welfare services that might otherwise develop economies and per capita income.

''But now it appears some in your White House are saying that family planning is not as important as development,'' Mr. Salim says with incredulity, referring to the draft statement leaked in June, proposing to reverse US priorities for family-planning aid.

If jobs are not found, the flow of emigrants, legal and illegal, to the US will grow, Western diplomats in Mexico City say.

Estimates of illegal Mexicans now in the US range from 2 million to 10 million. Congress is concerned enough to pass versions of the Simpson-Mazzoli bill making it illegal to hire such aliens.

* Crime. ''You see,'' says Marshall Green, a State Department consultant on population and a former ambassador, ''these are not cities like the ones we know.

''Half the people in them are under 16. They are restless and volatile. They don't go to school. They roam in gangs. They can't find jobs and they drift into crime.''

Police forces are not coping well enough in Lagos, Cairo, Bombay, Calcutta, or Delhi. The situation is ripe for exploitation: ''Extremism as well as crime takes advantage of slum overcrowding,'' says an adviser to President Hosni Mubarak in Cairo.

Sri Lanka, outwardly serene, sees a rising crime rate that Brig. Dennis Hupugalle attributes largely to immigrants from the countryside.

Jakarta and Rio are both seeing crime rates jump. Citizens in some areas have formed vigilante squads to protect themselves.

* Food. The so-called green revolution has taken hold in Asia, producing new strains or rice and other crops. Yet Africa, with poorer soil and primitive farming methods, lags behind. By 1980, third-world countries as a whole were spending $19.5 billion a year to import grain.

Cities in northeast Brazil, in the Andes countries of Bolivia, Peru, Chile, and Ecuador, in Central America, India, and the Mideast have all begun to see food supplies dwindle under the weight of numbers and badly managed farming.

Africa represents the most tragic prospect of all, worsened by the current sub-Saharan drought. Food supplies per head of population are falling, and UN figures show 145 million in 22 countries facing starvation.

In a private clinic 90 miles north of Nairobi, Magdalena Njeri worries: ''In my village,'' she says, ''people are hungry because of the drought. People like me see the problems of having too many children now.''

Mrs. Njeri has had eight children since 1966. She now takes contraceptive injections.

''If world population was increasing at 1 percent a year instead of almost 2 percent, there would still be ample margin for improving diets, as there was from 1950 to 1973,'' says Lester R. Brown of the Worldwatch Institute in Washington.

But food production is now about level with population growth. In Africa it is lagging behind.

What about food aid? It has been falling: US aid reached 15 million tons in 1965 (enough to feed 90 million people, Mr. Brown says). By 1982, however, it had dropped to 3.8 million tons.

What else can be done?

One answer is to step up information about family planning in urban slums.

Even in India, where 75 percent of people live in the country's 550,000 villages, Krishna Puri, the head of the New Delhi branch of the private Family Planning Association of India, says the aim now is to contact young urban mothers to show them how to space their children more widely.

Another method is to build new cities, or develop smaller ones, all with new jobs, to divert peasants from major cities. But this takes time.

Egypt has begun work on 10 new cities. The first one, called ''Tenth of Ramadan'' was started in 1977 some 30 miles outside of Cairo. After seven years of construction, progress is slow: only 5,500 people actually live there, according to chief engineer Hassan Rashidi. Most workers for the 80 factories now operating are driven to and from work by bus. Mr. Rashidi expects 150,000 will live in the city by 1988, and 500,000 by the year 2000.

In China, one plan would develop existing market cities of 200,000 people. And to locate more industries close to raw materials.

In India, some private industry is active: The huge Tata Iron and Steel Works in Jamshedpur, northern India, runs competitions between departments to see which can produce fewer children in one year. Program Chief Dhruva Dey says a trophy is to be awarded.

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