ESSAY-REVIEWS 649 



It must not be supposed, however, that the subject deals only with theorems 

 that require the attention of great mathematicians. The reader soon notices 

 some familiar questions, never out of date, whose antiquity will surprise 

 him. To mention only a few, 1669 is the date of the problem, " Anna took 

 to market 10 eggs, Barbara 30, Christina 50. Each sold a part of her eggs 

 at the same price per egg, and later sold the remainder at another price. 

 Each received the same total amount of money. How many did each sell 

 at first, and what were the two prices ? " 



An older one, due about 1220 to Leonardo Pisano, is to find a square 

 which, when either increased or decreased by 5, gives a square ; while a 

 still older type dates from a Chinese work of about the first century a.d., 

 giving a rule for determining a number having the remainders 2, 3, 2, when 

 divided by 3, 5, 7 respectively. 



Topics such as those previously mentioned have led to investigations and 

 results of the greatest importance, even in apparently unconnected subjects. 

 Considerable progress has been made by methods which are really arithmetical 

 in spirit, although apparently transcendental in character. At the present 

 time, however, equation (i) naturally suggests the study of functions defined 

 by the series 



^ (?) = 2 qlix, y,z . . .) , 

 or, again, 



X (s) - 2 [f{x. y.z . . )Y . 



where the summation refers to integer values for x, y, z . . . . 



Indeed, for many equations (i), the arithmetical development of the 

 theory has been carried furthest when the function (q) has been most 

 studied. Only in a few cases, such as, for example, Fermat's Last Theorem 

 or when / is an indefinite quadratic form, is it greatly in advance of the 

 analytical theory. Thus there is now no theoretical difficulty in finding the 

 number of solutions of 



(or even of 



f [x, y, z . . .) = m, 



where / is a definite quadratic form in n variables), as the function fp{q) 

 is a well-known modular function for which there exist other expressions 

 as a power series in q, and from which the required number of solutions at 

 once follows. 



When the properties of the function are not so well known, another 

 method has proved very successful lately in the hands of Hardy, Littlewood, 

 and Ramanujan. The unit circle is usually a line of essential singularities 

 for the function (q) . Taking the singularities given by e2Jr«a,6^ where 

 a and b are integers prime to each other, it is a simple matter to construct 

 a function !//■ (q) with these singularities, and so that the function <p{q) — ^ (q) 

 will not. have such heavy singularities as the original function </> (q) . The 

 question then arises. Is it likely that the coefficients of the expansion of !//• (q) 

 in ascending powers of q are an approximation, at any rate for large values 

 of m, to the corresponding coefficients in the expansion for (p{q) ? These 

 writers have established this possibility in a number of cases by means of 

 contour integration — a method which has already established the truth of 

 the famous prime number theorem. The first of two cases which we shall 

 mention, is to find an approximate formula for the number of partitions of m, 

 i.e. for the number of solutions in positive integers of 



X + 2y -{- sz + . . . = m, 



or the coefficient of q^ in the expansion of 



1/(1 -q){l- q') (I - q') {l - q*) . . , . 



