OF PttKSSUKE AND EQUILIBRIUM. 105 



U^yi- {^y + {jl)' }> the term Rdr of the equation 



(c) (252) will become a.3w, and the equation will become 

 O^XS^s-\->^.du, in which the coefficients of the variations 

 Sx, hjj ^z must be made to vanish separately ; so that it 

 affords three separate equations, which, however, are only 

 equivalent to two, since they contain the indeterminate 

 quantity a." Now, supposing the equation of the surface 

 u—0 to be r^ — x^. — y^ — z^ — q^ as in the sphere, the na- 

 tural sense of the symbol du is 2rdr—2x^x — 2y^y — 2z^z, 

 which must be =zO: but it must here be understood as rela- 

 ting only to the variations of jr, y, and z, exclusively of r, that 



is, §Mi= r-— Sj: + ~- ^y-\-.--dz. The subject may be further 



ha; by cz 



illustrated by an extract from the Mecanique Analytique of 

 Lagrange, Sect..ii. n. 7, 8. 



" Supposing, as is always allowable, the force P to 

 tend to a fixed centre at the distance jp, we havejpzzv' 



<(x — ay+(y — 6)2+(s— c)2>, and pdpzz(x — a)dx + 



(y—^)^y + (z — c)dz. Now, if ^ be perpendicular to a 

 given surface, its variation with respect to that surface will 

 vanish, and we have dp—0: the surface being spherical if 

 a, 6, and c, are constant, but of any other form when they are 

 considered as variable. If now the force P be in general 

 perpendicular to a surface represented by the equation adx 

 + ^dy + yds:z:0, in order to make it coincide with the equa- 

 tion (x — a)dx-\'(y — 6)dy + (s—c)ds=:0, which results from 



the supposition d/>=0, we must muke — = , and — = 



y z — c y 



V — ^ ^ 



— , whence x~-azz - (s — c), and y—h:=.- (z—c), and sub- 



2 — C y y 



stituting these values in the value of dp, it becomes dp= 



