ON THE THEORY OF THE EVOLUTE. 
477 
363] 
has the same value in the two curves respectively, or that, writing rri, n, S', k, t, l 
for the corresponding quantities in the second curve, then we have 
^ (m — 1) (m — 2) — 3 — k — \ (n — 1) (n — 2) — r — t, 
= i (m! -1) O' - 2) - S' - k’, =i (n - 1) O - 2) - r' - d ; 
and consequently that, knowing any two of the quantities m', n, S', k, r, i, the 
remainder of them can be determined by means of this relation and of Plticker’s 
equations. The theorem is applicable to the evolute according to the foregoing gene 
ralized definition ( J ); and starting from the values 
n' — m 2 — 23 — on — 6, 
/ / 
t = L , 
we find in the first instance 
t' = i (n' - 1) (n' - 2) - % (m - 1) O “ 2) + a + « -1'; 
and substituting in the equation 
m' = n (n — 1) — 2t — St, 
we find 
m' = 2 (n — 1) + (m — 1) (m — 2) — 28 — 2k — t ; 
and the equation t — k = 3 (n' — m) gives also 
k = - 3 (n' — m 1 ) + t, 
whence, attending to the value of n', we find the following system of equations for the 
singularities of the evolute, viz. 
n' = w 2 — 23 — 3/c — 6, 
m' = Sm( m — 1) — 63— 8k — 26— t, 
L —— L j 
k = 3m (2m — 3) — 123 — 1 ok — 3 6 — 21, 
and the values of t' and 3' may then also be found from the equations 
m = n! (n — 1) — 2t' — St , 
n 1 = m' (in — 1) — 23' — 3 k. 
I have given the system in the foregoing form, as better exhibiting the effect of 
the inflexions; but as each of the 0 contacts with the absolute gives an inflexion, we 
1 M. Clebsch in fact applies it to the evolute in the ordinary sense of the term, but by inadvertently 
assuming i' = k instead of t' = 0 he is led to some incorrect results.