106 Prof. Gauss on the Bepresentation of the Parts 



and we shall obtain an exactly similar expression for the co- 

 sine of the corresponding angle on the second surface by 

 changing a, b, c 9 a', #, d into A, B, C, A', B', Q. The two 

 expressions become clearly equal if the above-mentioned pro- 

 portionality takes place, and the second condition is already 

 comprehended in the first, as a little reflection will easily 

 show. 



The analytical expression of the condition of our problem 

 is, therefore, this: 



A«+B«-}-0 __ AA'+B.B'+C.C __ AA'+B'B'+C'G 

 a*+b*+c* aa'+bb'+ cc ><2_j_#«-|- c 'a * 



Let the value of these equal quantities, which must be a finite 

 function of t and u, be = m\ The quantity m is therefore 

 the index of the ratio in which linear quantities on the first 

 surface are increased or diminished in their representation 

 on the second surface (according as m is greater or smaller 

 than 1). This ratio will, generally speaking, be different in 

 different places : in the particular case in which m is constant, 

 there will be a perfect similarity also in the finite parts ; and if 

 m is besides = 1 , there will be a perfect equality, and the one 

 surface may be developed on the other. Putting for brevity 



(a 2 + b z + c 2 )df + 2(aa , + bb l +cc , )dt.du + (a! 2 + b n + d 2 )du 2 =a) 



we remark that the differential equation co = admits of two 

 integrations. Representing the trinomial oo as the product of 

 two factors linear with respect to d t and d u, either of the two 

 may be = 0, which will give two different integrations. One 

 of the integrations will be derived from the equation : = 



(a 2 + b z +c*)dt+{aa' + bb' + cJ+i\/[(a*+b*+c i )(a , *+b , * + 

 (?*)—{aa! + bV +cc**)]}du 



(where i is written for brevity instead of */ — 1, for it will be 

 easily seen that the irrational part of the expression must be- 

 come imaginary), the other integration will be the result of 

 a similar equation, which will be obtained by putting — i in 

 place of i in the former. If the integral of the first equation 



bethis p + iq= const. 



where p and q denote real functions of t and u, the other in- 

 tegral will be p _ iq _ const# 



It follows from this, that (dp + idq) (dp— idq) or dp* + dq* 

 must be a factor of co, or that 



w = n (dp* + dq 9 ) 

 where n is a finite function of t and u. 



Let us now denote by ft the trinomial into which dX*+ 



