of a given Surface on another given Surface. 107 



dY* + dZ* will be converted by substituting for dX, d Y, and 

 d Z their values expressed by T, U, d T and d U ; and let us 

 assume that in a similar manner, as before, the two integrals of 

 the equation i2=0 are as follow : 



P -f i Q = const. ; P— iQ = const, and 



SI = N. (rfP + rfQ 9 ) where P, Q, N denote real func- 

 tions of T and U. 



These integrations may evidently be effected (without taking 

 into consideration the general difficulties of integrating) be- 

 fore the solution of our principal problem. 



Now, if for T, U such functions of t and u are substituted 

 as will fulfill the condition of our principal problem, ft will be 

 changed into m 2 co, and we shall have 



(rfP + id Q) . (dP - idQ) m*n 



(dp +idq) • (dp — idq) N 



But it will be easily seen that the numerator in the first part 

 of this equation cannot be divisible by the denominator, ex- 

 cept if either 



dP + idQ is divisible by dp + idq, and d¥—idQ by dp—idq 9 



or, 

 dF + idQ is divisible by dp*~idq, and dP— idQ by dp-\-idq. 



In the former case dV + idQ will therefore vanish if dp -f 

 idq = 0, or P + * Q will be constant if p + iq is supposed to 

 be constant; that is to say, V + iQ will be a function of p-\-iq 

 only, and in the same manner P — i Q will be a function of p — i q. 

 In the latter case P + zQ will be a function of p— iq, and 

 P— z'Q a function of p + iq. It is easy to perceive that the 

 reverse of these positions likewise holds good, or that if for 

 P + fQandP — iQ functions o$ p + iq or p—iq (either re- 

 spectively or inversely) are assumed the divisibility of SI by co, 

 and consequently the above required proportionality will take 

 place. 



It will easily be conceived that if, for example, we suppose 



P + **Q =f(p + iq), P-/Q =f(pe-iq) 

 the nature of the function f is already given by that off. For 

 if among the constant quantities which it involves, there are 

 none but real quantities, the function f 1 must be identical with 

 f; in order that real values of P and Q may correspond to 

 real values of p and q : in the contrary case, f will only be 

 distinguished fromy by having in the imaginary quantities 

 which f involves — i instead of -fz*. 



We have next, P = \f{ p + iq) + if'( p — iq) 



iQ = if(p+iq)-U\p- i 9)> 



or, which is the same, as the function f is assumed quite ar- 



P 2 bitrarily 



