108 Prof. Gauss on the Representation of the Parts 



bitrarily (involving at pleasure constant imaginary quantities), 

 P is equal to the real, and z'Q (or — *Q in the second solution) 

 equal to the imaginary part o?f(p + iq) 9 and by elimination 

 T and U will be represented as functions of t and u. Thus 

 the proposed problem is solved quite generally and com- 

 pletely. 



6. If we represent any determinate function of p + iq by 

 p* + * 4 (where $ and q' are real functions ofp and q\ it will 

 be easily seen that likewise the equations 



p ] + iq* = const, and p ] — iq 1 = const, 

 will represent the integrals of the differential equation w = ; 

 indeed, these equations will respectively agree with the above 



p + iq = const, and/?— iq = const. 

 In like manner the integrals of the differential equation /2=0, 

 viz. P' + *Q' = const, and F— iQ = const, 



will agree with the above, 



P + /Q = const, and P — iQ = const, 

 if P'-f I Qf represents any determinate function of P + z'Q 

 (while F and Q' are real functions of P and Q). Hence it is 

 clear that in the general solution of our problem which we 

 have given in the preceding article, p' and q' may be substi- 

 tuted for p and q, and P' and Q' for P and Q. Although this 

 change does not add to the generality of the solution, yet in 

 practice one form may be more applicable to one, and another 

 to another purpose. 



7. If the functions arising from the differentiation of the 

 arbitrary function/* and/ 7 are denoted respectively by <p and 

 $', so that d .f'v = $ v . d v and d .f ' v = <p'v .dv.we shall have 

 in conformity with our general solution 



therefore, *£ = *(p + iq).*(p-iq). 



The scale of linear dimensions is determined by 



8. We shall now illustrate our general solution by some 

 examples by which the manner of applying it, as well as the 

 nature of some circumstances which may come into considera- 

 tion, may be best explained. 



Let the two surfaces be in the first place planes, in which 

 case we may put 



x =■ t y y = w, z = 

 X = T, Y = U, Z = 0. 



The 



