54 



a = IX +jy + kz, a = ix +jy' + kz, ... (72) 



where xyz are the rectangular co-ordinates of m, and ijk are the 

 threeoriginal and fundamental symbols of the present Calculus, 

 denoting generally three rectangular vector-units, and subject 

 to the laws of symbolical combination which were communi- 

 cated to the Academy by the author in 1843, and are included 

 in the formula (4) of the present Abstract, And then, by 

 (35), the coefficients of the cubic equation (56) will take the 

 following forms, which easily admit of being interpreted, or 

 of being translated into geometrical enunciations : 



n' = S . m(a;^ + 2/^ + 2^) ; "^ 



w'2 = S . mm [ {yz - zy'f + {zx - xz'f + {xy - yx'Y' j ; >0^) 



n"^ = S . nim'm" { (yz - zy')x" + (zx - xz')y" + (xy - yx)z ^^.j 



In fact, the first of these three expressions is evidently the 

 sum of the three quantities (69); and it is riot difficult to 

 prove that, under the conditions (71), the second expression 

 (73) is equal to the sum of the three binary products of those 

 three quantities; and that the third expression (73) is equal 

 to their continued or ternary product: in such manner as to 



give 



5i + S2 + S3 = %^ ; "^ 



5lS2+S2S3 + «3«l = '^'^; \ (74) 



Perhaps, however, it may not have been noticed before, that 

 expressions possessing so internal a character as do these 

 three expressions (73), and admitting of such simple interpre- 

 tations as they do, without any previous reference to the axes 

 of inertia, or indeed to any axes (since all is seen to depend 

 on the masses and mutual distances of the several points or 

 elements of the system), are the coefficients of a cubic equation 

 which has the well-known sums, S . mx^, S . ony'^, S . mz^, re- 

 ferred to the three principal planes, for its three roots. In the 

 method of the present communication, those expressions (73), 



