of Edinburgh, Session 1885 - 86 . 
577 
cases to enable one readily to see what change would be necessary 
in applying it to the next case, Notwithstanding these drawbacks, 
however, there can be no doubt that if any one name is to be 
attached to the theorem it should be that of Laplace. 
The sum of his contributions may be put as follows : — 
(1) A proof of the theorem regarding the effect of the transposi- 
tion of two adjacent letters in any of the new functions. (xii. 2) 
(2) A mode of arriving at the known solution of a set of simul- 
taneous linear equations. (xm. 2) 
(3) The name resultant for the new functions. (xv.) 
(4) A notation for a resultant, e.g. ( x a. 2 6. 3 c) . (vxi. 2) 
(5) A rule for expressing a resultant as an aggregate of terms 
composed of factors which are themselves resultants. (xiv. 2) 
(6) A mode of finding the number of terms in this aggregate. 
(XVI.) 
LAGBANGE (1773). 
[Nouvelle solution du probleme du mouvement de rotation d’un corps 
de figure quelconque qui n’est anime par aucune force acc61era- 
trice. Nouv. Mem. de V Acad. Roy (de Berlin). Ann. 
1773 (pp. 85-120).] 
The position of Lagrange in regard to the advancement of the 
subject is quite different from that of any of the preceding mathe- 
maticians. All of those were explicitly dealing with the problem of 
elimination, and therefore directly with the functions afterwards 
known as determinants. Lagrange’s work, on the other hand, consists 
of a number of incidentally obtained algebraical identities which we 
now-a-days with more or less readiness recognise as relations between 
functions of the kind referred to, but which unfortunately Lagrange 
himself did not view in this light, and consequently left behind him 
as isolated instances. With him x, y , z and x', y' , z' and x", y n , z" 
occur primarily as co-ordinates of points in space, and not as coeffi- 
cients in a triad of linear equations ; so that 
( xy'z " + yz'x" + zx'y" - xzy" - yx'z" - zy'x ") , 
when it does make its appearance, comes as representing six times 
the bulk of a triangular pyramid and not as the result of an elimi- 
nation. In days when space of four dimensions was less attempted 
to be thought about than at present, this circumstance might pos- 
2 Q 
VOL. XIII. 
