48 
[171 
171. 
NOTE ON MR SALMON’S EQUATION OF THE ORTHOTOMIC 
CIRCLE. 
[From the Quarterly Mathematical Journal, vol. I. (1857), pp. 242—244.] 
Let U 1 = 0, U. 2 = 0, U 3 = 0 be the equations of three circles, and let V be the 
functional determinant of U 1 , U 2 , U 3 , the functions being in the first instance made 
homogeneous by the introduction of a variable z, which is ultimately replaced by 
unity; then the equation of the circle cutting at right angles the three given circles, 
or, as it may be called, the orthotomic circle, is V = 0. This elegant theorem of 
Mr Salmon’s is connected with the theory developed by Hesse in the memoir, “ Ueber 
die Wendepuncte der Curven dritter Ordnung,” Crelle, t. xxvm. (1844), p. 97). 
In fact, let U 1 = 0, U. 2 = 0, U 3 = 0 be the equations of three conics, the locus of 
a point such that its polars with respect to each of these conics, or indeed with 
respect to any conic having for its equation \U 1 +yU 2 +vU 3 = 0 (where X, y, v are 
arbitrary), pass through the same point, is a curve of the third order V = 0, where 
V is the functional determinant of U 1 , U 2 , U 3 . 
Conversely, if the curve of the third order F=0 be given, and U be a function 
of the third order, such that the functional determinant of , or, what 
ax ay dz 
is the same thing, the “ Hessian ”. of the function U is equal to V, a condition 
which may be written V= H( U), then we may take for the conics any three conics 
dU dU dU 
the equations of which are of the form X 
dx 
- + y + v = 0. The equation V=H(U) 
affords the means of determining U; in fact, we shall have U=aV+bH(V), where a and b 
are constants to be determined. This gives H{U)=H (aV+bH (V)')=AV+BH(V), where 
A and B are given functions of a, b (a practical method of determining these functions 
was first given in Aronhold’s memoir, “Zur Theorie der homogenen Functionen dritten 
Grades von zwei Variabein,” Crelle, t. xxxix. (1850), pp. 140—159); and we have therefore