in tJte Position of a Point in ISpace. 



135 



(Y„=i<), ^^-3=0. If the coordinate axes through this origin are taken 

 to coincide with the free axes of the system of masses L, we have 

 also /j—O, 72=0, 73=0, By considerations like those employed at 

 Analyst, ix, pp. 38 to 40, it will appear that formulas (46) and (51) 

 still hold good, the constants a, |5, y, etc., referring to the new axes 

 with the same meanings as before. Let us neglect second diflFeren- 

 tials oi w in comparison with /« in the expression for V in (51). This 

 gives V = /r. Also let n be written instead of n + \. Then giving to 

 ^,,y, k their equivalents from (48), we reduce (51) to 



l\d.w-U7^,-^/j,)d;w-Uf^^-^p,)d;w-U^ 



y (52) 



— {^^ft)d/l,;w — {i]^^ P,)d^d.w - (//„H- fJ,)d,d,w 

 If we write 



—zdz 



VpjiW 



A =nfi^{d>Y, B =np,{dyy, C =nf3,{dz)\ 



Aj= w^j(c?,(')3^ A.^z=.ni]^d.i'{dyY, A^z=.ni]^d.v{dzY, 



B^=7ij]^{d.>'Ydy, Ba= n6.^{dy)'^, B^= mf^dy{dz)^', 



C ^=nr/2 {d.r) 2 dz, C 0= n?/^ {dy) ^ dz, L\ = n6^{dz)^, 

 E ^ ndd.rdydz, 



(52) may be put in the form 



, A /d'w\ , , /d'iv\ , . /d'w\ ^/d'w\ ,Jdhc\ ] 



H7i?)+^''{w)^^''\'d^)^^\d^^^^ 



r^( d''w\_ ./dw\ 

 \dydz/ \d,v/ 



^ / dSo \ ^/dw\ 

 ,^ / d„'w \ ^/dw\ 



W53) 



"t (54) 



