558 
[553 
553. 
TWO SMITH’S PRIZE DISSERTATIONS. 
[From the Messenger of Mathematics, vol. n. (1873), pp. 161—166.] 
Weite dissertations : 
1. On the condition of the similarity of two dynamical systems. 
2. On orthogonal surfaces. 
1. We may consider two particles m, m, describing similarly two similar paths 
(which for convenience may be taken to be similarly situate in regard to two sets of 
rectangular axes respectively), viz. this means that the times t, t' of passage through 
s' t' 
corresponding arcs s, s' are proportional. The ratios -, , are thus each of them 
S b 
constant; and this must also be the case with the ratio V , of the velocities v, v' at 
v 
s' v' t' 
corresponding points; since it is clear that we must have - = 
s v t 
Now in order that the two particles may move as above under the action of 
any forces upon the two particles respectively, it is clearly necessary that the forces 
F, F' at corresponding points shall act in the same direction, and be in a constant 
ratio of magnitude. To obtain this ratio, imagine the two particles, masses m, m!, 
moving as above, in the corresponding infinitesimal elements of time t, t, with the 
velocities v, v through the infinitesimal arcs a, a respectively, ; 
\t t v v ’ a s) 
the deflections from the tangent will be ^ — r 2 , | ~ t' 2 respectively, and these must 
be in the ratio of the corresponding arcs a, a, viz. we must have 
F' T ' 2