226 REPORT—1862, 
P:Pi:lirit:tt= BC —F?— (B4C)u4+w, 
: C'A'—G? — (C'+A')u4+wye 
: A'B'—H?— (A'+ B)u4+wv’, 
: GH'—A'F’ +F' u, 
: H'F’—BG' +G' u, 
: P'G'—C'H' +H'u, 
or putting therein F’=0, 
@i:F:lirrt:¢= BO —(B'+C')u+uv' 
: CA'—G?—(C'+A')u+uv? 
>: A'BI—H?—(A'+ But? 
3) (GH! 
; —B'G'+G'u 
; —C’H'+H'u 
by means of which Segner’s equations may be verified. I have given this 
analysis, as the first solution of such a problem is a matter of interest. 
148. There is little if anything added to Segner’s results by the memoir, 
Euler, ‘‘ Recherches sur la Connaissance Mécanique des Corps” (1758), which 
is introductory to the immediately following one on Rotation. 
149. Relating to the theory of principal axes we have Binet’s “‘ Mémoire 
sur les Axes Conjugués,” &c. (1813). The author proposes to make known 
the new systems of axes which he calls conjugate awes, which, when they are 
at right angles to each other, coincide with the principal axes; viz. consider- 
ing the sum of the molecules each into its distance from a plane, such distance 
-being measured in the direction of a line, then (the direction of the line being 
given) of all the planes which pass through a given point, there is one for 
which the sum in question is a minimum, and this plane is said to be con- 
jugate to the given line, and from the notion of a line and conjugate plane 
he passes to that of a system of conjugate axes. The investigation (which 
is throughout an elegant one) is conducted analytically; the coordinates 
made use of are oblique ones, and the formule are thus rendered more com- 
plicated than they would otherwise haye been; in referring to them it will 
be conyenient to make the axes rectangular. 
150. One of the results is the well-known equation 
(A’—0)(B'—e)(C —0)—F"(A’—0) —G"(B' 0) —H(C' —0) + 2F G'H'=0; 
which, if @,, y,, 2, are the principal axes, has for its-roots fw,*dm, [y,’dm, 
zdm. 
And the equations (1), p. 49, taking therein the original axes as rect- 
angular, are , 
K' 
(s— 3) cosa+ ®' cosp+ 6’ cos y=0, 
4B’. cosa+ (38'—5; Joos B+ FF cos y=0, 
+6' cosa+ df’ cos p+ (€'—5) cos y=0, 
where @', 15’, €’, df’, @', ®' denote the reciprocal coefficients @'=B'C’—F? 
