[624 
of x, y, the 
625] 
587 
) = 0, 
625. 
ON THE CONDITION FOR THE EXISTENCE OF A SURFACE 
CUTTING AT RIGHT ANGLES A GIVEN SET OF LINES. 
[From the Proceedings of tice London Mathematical Society, vol. vm. (1876—1877), 
pp. 53—57. Read December 14, 1876.] 
In a congruency or doubly infinite system of right lines, the direction-cosines 
a, /3, 7 of the line through any given point (x, y, z), are expressible as functions 
of x, y, z; and it was shown by Sir W. R. Hamilton in a very elegant manner 
that, in order to the existence of a surface (or, what is the same thing, a set of 
parallel surfaces) cutting the lines at right angles, adx -f fidy + ydz must be an exact 
differential: when this is so, writing V = J (adx + ¡3dy + 7dz), we have V = c, the 
equation of the system of parallel surfaces each cutting the given lines at right angles. 
The proof is as follows:—If the surface exists, its differential equation is 
adx + ¡3dy + ydz = 0, and this equation must therefore be integrable by a factor. 
Now the functions a, ¡3, 7 are such that a 2 + /3 2 + 7 2 =l, and they besides satisfy a 
system of partial differential equations which Hamilton deduces from the geometrical 
;os G+isin C, notion of a congruency; viz. passing from the point (x, y, z) to the consecutive 
point on the line, that is, to the point whose coordinates are x + pa, y + p/3, z+py 
(p infinitesimal), the line belonging to this point is the original line; and conse 
quently a, ¡3, 7, considered as functions of x, y, z, must remain unaltered when these 
variables are changed into x + pa, y + p/3, z + py, respectively. We thus obtain the 
equations 
joint (b cos G, 
ire the points 
formulae give 
74—2 
da da da 
*T* + l3 dy +r <d, =°- 
d/3 n d(3 d/3 A 
a £ +0 dy + ^£ =0 ’ 
a p + 0 p + y p„ 0. 
dx dy dz