328 report — 1865. 



These expressions of u and it' may be verified by a comparison of their 

 general factors with the general factors in the formulas (26) : for some of 

 them, this comparison requires the Eulerian identity already cited (E). Limits 

 of n and 2 are 1,+ oo , and — oo , +co ; the transformation of the products 

 into sums is effected by means of (7) . 



If a>=a+bi, and if the positive quantity b increases without limit, a re- 

 maining finite, we infer, from (26), that 



lim yj, (a+bi) = + l, hm&t^cos^ + mn^. 



/— _ o o 



V2e~ a 



"We shall presently see that </>(o»)=»//{ J; hence if u)=j and b increase 



without limit, lim (p(l\ = lim \p (bi) = -f 1, lim \p ( - J = lim <p (ib) = 0, 



V2e~8 



The principal properties of ((f)) w and \|/ (w) are deducible from the Theory of 

 the Transformation of Elliptic Functions. The general problem considered in 



that theory is " Given w=^~— — , where a, b, c, d are positive or negative in- 



a-\-b£l 



tegral numbers, to express the Theta functions containing £2 by means of the 



Theta functions containing w." The determinant ad— he must be different 



from zero and positive, because the coefficients of i in the imaginary parts of 



o) and SI must both be different from zero and positive ; if ad — 5c=n, the 



transformation is said to be of order n. Let A, A', X, X', v, v' be the same 



A' K' 



functions of £1 that K, K', k, k, u, «' are of w ; since H=i — , u=i — , the 



A K 



equation to=— — implies the existence of two others of the form 

 ^ a + ba L 



Ik =aA + MA',' 



(29) 



2-iK'=cA + diA'; 

 M 



in which M is a coefficient termed the multiplier; when A has been found, 

 M is determined by the equation 



L=^(a+bQ)=±(e+da); (30) 



it also satisfies the relation 



M»_lMlz^ —* (31) 



n K (l-K 2 ) 'd\ 



If n=l, the theory of the transformations of the first order has been com- 

 prised by M. Hermite in the single formulaf, 



* Fundanienta Nova, p. 75. 



t Liouville, New Series, vol. iii. p. 26 ; and, with less detail, in the Comptes Eendus, 

 vol. xlvi. p. 171. 



