88 Proceedings of the Royal Irish Academy. 



which by the conditions (A) is equal to the cosine of the angle between the 

 normals to Q = const, and R = const., and to the cosine of the angle between 

 the normals to B = const, and P = const. 



Moreover, the cosine of the angle between the normal to P = const, and 

 the line x = y = z is 



dP dP dP 

 t/ 3 dx dy dz 



o L'dPy fdP\* fdP\ 2 



A*) + W + w 



which is equal to the similar expressions formed with the partial differentials 

 of Q and R ; hence, the three normals make the same angles with the line 

 x = y = s. 



Similar results hold when the axes are oblique, provided they make equal 

 angles with one another. 



IV. If we write 



f(u)du - j (P, + P,0 + . . . P n 9*- l )(dx r + 0dz 2 + ... 6 ,l - 1 dx„), 



P,dx l + K dx„ +... P n dx- : + 6( ) + ... 6"- 1 ( P x dx n + ■■■ PJ*> ) ) , 



= 1' 



we see at once thai the conditions 



dP m dPm+k 



dXfi rf.'vu- 



are the conditions that the term independent of 6 under the integral sign, 

 and the coefficients of the several powers of f) shall be perfect differentials. 



It follows at once that in general the integral has a sense, provided that 

 nowhere in the path of integration do the "P"s become infinite. 



If the function to be integrated were of the form -*- > where <j> (n) is 



finite for all finite values of u, and if we write 



u - vl 



X, + A,0 + . . . X n 9 n - 1 



we shall find on solving for the " X"s from the equations arising from 



(u) ^ P t + P,6 + . . . P n 9"- 1 = (A, + X a + . • • A,^"- 1 )^, - *l + {x, - x/) 



+ . . . 0'- 1 (x„ - x„')) 



