be 
ON THE RECENT PROGRESS OF ANALYSIS. 53 
in his earlier essay, he assumes y equal to a rational function of z, whose 
coefficients are elliptic functions, and shows that this assumption satisfies the 
_ differential equation already mentioned. It may be asked what is gained by 
the introduction of elliptic functions into a problem of which, as we have 
seen, particular cases (e.g. the transformations of the third and fifth order) 
ean be solved by algebraical considerations. The answer is, that the pro- 
_ perties of these functions enable us to transform the assumed relation between 
_ yand z in a manner which would otherwise be impracticable. It is con- 
ceivable that any particular case might be solved by mere algebra, but it 
does not seem possible to discover in this way a general theorem for trans- 
formations of all orders, and practically the labour of obtaining the formule 
for the transformation of any high order would be intolerable. 
Having proved the theorem for transformation in nearly the same manner 
as he had already done, M. Jacobi developes the demonstration which, as 
‘we have said, Legendre hinted at in No. 130 of Schumacher’s Journal. 
He then proceeds to consider the various transformations of any given 
order. We have seen that the modular equation for those of the third order 
rises to the fourth degree, that is to say, for a given value of the modulus of 
the original integral four new moduli exist, corresponding to four new in- 
tegrals, into which the given one may be transformed. These four trans- 
formations are all included in the general formula for the third order; but 
it is to be remarked that in general only two of the roots of the modular 
equation are real. Thus there are two real transformations and no more. 
The same thing is true, mutatis mutandis, of the transformations of any 
prime order (to which M. Jacobi’s attention is chiefly directed), that is to 
say, there will be 2 + 1 transformations of the mth order, » —1 of which 
_ are imaginary. The two real transformations are called the first and the se- 
cond ; the second is sometimes called the impossible transformation, because 
it presents itself in an imaginary form*. Of the formule connected with 
these two transformations M. Jacobi gives copious tables. 
He next shows, in a very remarkable manner, that, corresponding to a 
transformation in which we pass from a modulus & to a modulus A, there 
exists another, whose formulz are derivable from those of the former, in 
__ which we pass from a modulus 1 —k to a modulus 1 — a’, or which 
connects moduli complementary to A and k. The latter is accordingly called, 
_ with reference to the former, the complementary transformation. The first 
real transformation of & corresponds to the second real transformation 
Pi V1 —k?, and vice versd. 
_ _ The next theorem which M. Jacobi demonstrates is not less remarkable. 
_ It is that the combination of the first and second real transformations gives 
by a formula for the multiplication of the original integral, or, in other words, 
_ that the modulus of the integral which results from this double transforma- 
tion is the same as that of the original integral, so that the two integrals 
& differ only in their amplitudes. Of this theorem he had in the earlier part 
of the work proved some particular casest. 
regs, 
en! 
_ * Mr. Bronwin, in the Cambridge Mathematical Journal and in the Phil. Mag., has 
"made some objections to this transformation ; but from a correspondence which I have re- 
- cently had with him, I believe I am justified in stating that he does not object either to M. 
 Jacobi’s result or to the logical correctness of his reasoning, but only to the form in which 
the result is exhibited. 
+ It may be shown that if we pass from / to a by the first transformation, we can pass 
from Vi— 2 to V1 — # also by the first transformation. Also, as has been said, we 
_ derive from the transformation {% to a} a transformation { VI —FPto V1— a2}, and 
