22 
PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
and 
6 V or A ^ ^ 
5yS3 ■ ^ 5^3 ’ 
so that, if Zq and [dz/dx)Q are known, is known. 
The differential equation (30) which we have to satisfy is 
= -z\ 
or 
A, Y ^ / cip " \\l ■ 
Now, by expansion. 
2 (^J = 2 + +S[^,+ P + 3 [.4, + 4-4D. - f 
“ +SlA, + i (A,A, + iA/) - iA,U, + A^i*] f‘ 
+ 3 [^5 + i 3 ) — } (A^-Aq + ~ T 2 A^ 1 ^] 
+ 3 [Aq + ^ (A^A^ + A^A^ + ^Ag^) — g- (A^^A^ + 2AjA2Ag + ^A^^) 
+ i4- (A,^Ag + §A,^A./) - rhA^A^ + 
+ 3 [A^ + ^(A^Ag-j- A^Ag-j-AgA^) — ^(Aj^^AgA-^Aj^A^A^-i-A^Ag^A- A^^Ag) 
4" TA (Ai^A^ + SA^^A^Ag + Aj^A^^) — xIf (Ai^Ag + 2^4 
+ 2io^l^^2-T*4A']F+.(31) 
And 
^^'+‘TiY»("/"»>=2 + (3-2z + f2.l)f+(4.3.4‘ + h3.2'^ + 5.2.l)f 
A, 
A, 
A^ 
+ 0-4i: + f4-3i; + ,1.3.2^’+i.2.l)f3 
+ (6.5^-«+i.5.4^; + ^i4.3^*+4.3.2;^ + i.2.l)f* 
A, 
A 
A. 
4 „ . yR . 6 _ . Hj , 4 ^ ^ A, , 1 ^ 
+.(32) 
By equating the coefficients in (31) and (32) we are able to determine the A’s. The 
law of the series (32) is obvious, and sufficient of the series (31) is written down to 
enable us to find.y4g. We can, however, obtain a good approximation to higher 
coefficients, because the coefficients in (31) become relatively unimportant. 
