16 SIR G. H. DARWIN: FURTHER CONSIDERATION OF THE 
§ 4. Note on §§ 16, 17. The Integrals cr 2 , cr 4 , £ 4 and o>/, pi, </>/. 
An improvement has been made in the method of computing all these. The 
functions to be integrated were written in every case with a common factor 
cosec 2 y (A^—Fj 2 ) ; now this is equal to k 2 cos 2 9+ k ' 2 sin 2 </>. In consequence of the 
substitution of this value for the common factor we are able to obtain the result as 
the sum of, instead of the difference between, two numbers. Another consequence is 
that we can dispense with the series of functions denoted A_ 2 and fi_ 2 . 
A single example of the way in which this change is applied will suffice. If we 
write 
/(A 2 ,») = «»A 2 ,»-^»A 2 . 2 + r »A 2 ,‘-3»A ! , 6 'l ^ 
>n — 1 , 2 , 
/(A a „ 2 ) = «»A 2 /- / 3' l A 2 / + y»A 2 , s -8"A 2 „ 8 J 
and denote by f(Cl 2n °), f(L1 2u 2 ), corresponding functions with ct !, y, S' for a, /3 , y, S 
and O in place of °A; it is easy to show that 
<r 2 = ® COS 2 /3 COS 2 y { K- [/(A 2 2 )/(n i 0 ) +/(A/)/(fl 2 a ) - <?/(A 2 2 )/(n 2 °)] 
77 
+ ff 2 [/(A 2 °)/(fi4 2 )+/(A 4 0 )/(O 2 2 )-^/(A 2 °)/(fi 2 2 )]}. 
The computations as revised gave 
o - 2 = *0136760, £4 = -000092343, cr 4 = -000176218. 
Taking log = 9-5461687, I found 
^[i(o - 2 ) 2 + 2£ 4 ]-icr 4 = --00050051. 
This differs by 4 in the seventh place of decimals from the old value. 
In evaluating off, pi, <£/ when we use the form of involving f 0 , f 2 , J\, &c., we 
have to put 
[Ps (/x )] 2 = F 0 sin 6 6—F 2 sin 1 0 cos 2 6+ F 4 sin 2 0 cos 2 6, 
and F 0 , F s , F± are easily found from «, /3, y, S. 
I then write 
Lo =f>F 0 , L 2 — f 0 F 2 +f 2 F 0 , Z 4 =f 0 F i +f 2 F 2 +f i F 0 , &c., 
and, when i denotes the order of the harmonic S/ concerned, write 
/( A 2 „°) = L 0 i+6 A 2n °-L 2 i+4 A 2 « 2 +^ 4 i+2 A 2re 4 - ...(n = 1, 2), 
/(A 2;i 2 ) = L 0 i+G A 2n 2 — L 2 i+i A 2n i + L 4 i+2 A 2n 6 — ...(n = 1, 2). 
These functions are then combined to give the required integrals. 
