48 REPORT—1846. 
tions of x, the differential equation will be satisfied, provided U and V fulfil 
two general conditions, the second of which is found to be deducible from 
the first. He then makes an assumption which is equivalent to assigning 
particular forms to U and V, and thence shows, by a most ingenious method, 
that these forms of U and V are such as to fulfil the first of the required condi- 
tions, which, as has been said, implies the other. He thus verifies, @ poste- 
riori, the assumed value of the function y. 
In proving that the forms assigned for U and V have the required pro- 
perty, it is necessary to pass from an expression of the value of 1—y in terms 
of x to one of 1— Ay in terms of the same quantity. This is done by 
means of a remarkable property of the functions U and V, namely, that if 
1 
han 
justed) become ne or G Therefore, in any form in which the relation con- 
necting y and 2 can be put, we may replace x by i? provided we at the 
in both x be replaced by - or y will (the constants being properly ad- 
same time replace y by wh This has been called the principle of double 
substitution, and by means of it we pass from the expression of 1—y to that 
of 1'— ~, and thence obtain that of 1— Ay. It is to be observed that 
this principle is used merely to prove a certain property of the functions 
U and V. Of course, as the change of 2 into Pm implies that of y into = 
in the finite relation between these quantities, the same thing will be true in 
the differential equation by which they are connected, a remark which may 
very easily be verified. But, on the other hand, it by no means follows that 
because it is true in the differential equation therefore any assumed finite 
relation between y and x having this property is the integral required. The 
property in question therefore does not enable us to verify any assumed 
value of y. . 
This remark has reference to a communication from Legendre which ap- 
pears in No. 130 of Schumacher’s Nachrichten, vi. p. 201. In it he gives 
an account of M. Jacobi’s researches, and an outline of the demonstration of 
which we have been speaking. I find it impossible to avoid the conclusion 
that this great mathematician mistook the character of the demonstration in 
question, and that to him it appeared to be in effect a mere verification of 
the assumed value of y by means of the principle of double substitution. 
He remarks that the direct substitution of the value of y in the differential 
equation is impracticable, but that M. Jacobi had avoided this substitution 
by means of “ une propriété particuliére de cette équation qui doit étre com- 
mune aux intégrales qui la représentent.” This property is the principle of 
double substitution ; and after showing that it is true of the differential 
equation, the writer proceeds thus: “‘Ce principe une fois posé, rien n'est 
plus facile que de vérifier ’équation trouvée y= y? car par la double sub- 
stitution on obtient la méme valeur de y 4 un coefficient prés qui doit étre 
égal a l'unité;” and, after a remark to our present purpose immaterial, con- 
amie : | Oar 
cludes, “ Ainsi se trouve démontrée généralement l’équation y = 7 aint 
que, etc.” 
As we have seen, such a verification would be wholly inconclusive, nor is 
