MR. A. R. FORSYTH OH A CLASS OF FUNCTIONAL INVARIANTS. 
73 
in character from tlie differential invariants of M. Halphen and the ternary recipro- 
cants of Mr. Elliott. 
The earliest record of M. Halphen’s investigations is his well-known thesis, * 
wherein he considers the invariance of a differential equation /{x, y, y', y", . . . ) = 0, 
when the single independent variable x and the single dependent variable y are 
(p. 20, loG. cit.) subjected to the transformation 
X _ Y _ I 
ax H- by -p c a'x h'y + c' a!'x -f b"y -h c" 
The only reference in the thesis to the case of three variables is (p. 60) in the 
concluding paragraph, where it is said that the theory can be extended to the case of 
one dependent variable 2 and two dependent variables x and y, the transformation 
suggested, but not explicitly stated, being 
X _ _Y__Z_ 1 _ 
ax + ySy -f- 7 ^ -f- 8 ax + j3'y + y'z 8' a'x -f 13"y + y"z -p 8" a"x -H ^'"y -f- <y'''z + S'" 
M. Halphen, again, t considers differential invariants, in which the last trans¬ 
formation is effected on functions of the three variables ; but in this investigation 
y and 2; are taken to be two dependent variables of the single independent variable x. 
Mr. Elliott’s theory | of ternary reciprocants is closely connected with the con¬ 
cluding paragraph of M. Halphen’s thesis; the functions are invariantive for inter¬ 
changes of z, X, y, where z is a variable dependent on x and y ; and the pure reciprocants 
are invariantive for the above-suggested transformations. 
The theory in this memoir deals almost entirely with the case of three variables, 
z, X, y, where z is a dependent variable, and x and y are independent variables. The 
transformations, through which the invariance is maintained, refer to the independent 
variables only ; they are— 
x y 1 
«i -I- yS^X + 7 iY + AX + «3 + AX -t 73 Y 
The dependent variable is left untransformed; it does not enter into the equations of 
transformation. 
It follows, from the difference between the transformation in the theory here 
other classes, such as the differential invariants of M. Halphen and the ternary reciprocants of Mr. 
Elliott. 
* ‘ Sur les Invariants Differentiels,’ Paris, 1878. 
t “ Sur les Invariants Differentiels des Courbes gauches,” ‘ Journ. de I’Ecole Polytechnique,' vol. 28’ 
1880, pp. 1-102. 
t “On Ternary and n-ary Reciprocants,” ‘London Math. Soc. Proc.’ vol. 17 (1886), pp. 171-196; 
“ On the Linear Partial Differential Equations satisfied by pure Ternary Reciprocants,” ibid., vol. 18 
(1887), pp. 142—164; “On pui-e Ternary Reciprocants and Functions allied to them,” ibid., vol. 19 
(1888), pp. 6-23. 
MDCCCLXXXIX. —A. 
L 
