NUMBER OF PARTITIONS OF WHICH A GIVEN NUMBER IS SUSCEPTIBLE. 419 
putation of tlte coefficients, we find 
9 fM0-10.r+ 9. 10^-1 + 20. 10^-2+33. 10^-3+48. 10x_4- 15. 10^-5-16. 10.,_6-15.10^_7-12.10.,-8-7A0^_9} 
2880\ + 2^._i{ 0. 10^- 1.10^_i + 0. 10^_2+3. 10^_3+8.10^_4+ 15. 10^_5 + 24. 10^-6 + 35. 10^_7 + 48. 10^-8 + 63 . 10^_ 
_ 9 
2880 
0. 10^- I.IO4;-, +20. 10,._2+ 3. 10^_3 + 48. 10^-4+15. 10^-5- 16. 10^_6 + 35.10^_7-12.10^,-8+63.10^_9|. 
(39.) Finally, we have to consider the portion Z'" of Z originating in Q^i, in which 
the values of X(» %ij &c. are given by the equation 
>:. = l44i-ll-12,+0.12,_,-|-5.12,_,-20.12,_3 
4.12,. 
the coefficients being those of equation (51.) in their order of circulation. We have 
also, since in this case 771 = 12, ^=5, and therefore prime to each other, 7;=1, ^=12, 
m= 60. Whence 
KiO '^x~^ ”1~ 'Xjm—l^x—m + \ “H %1 H~ • • •%)«—! 
144 ’ 
and therefore 
26.r 26 
1-60.-59} 
- 11 . 60 .+ 0 . 60.-1 + 5.60.-2-20. 60.-3-27. 60.-4} 
2880 L - J 
-36. 60.+5. 60.-1 + 16. 6._2-27.60._3-4. 60.-4-11. 60.-5 + 0.60.-6 + 5. 60.-7-20. 6O._8-27.6O._9 } 
^ + 5 . 60 .- 20 . 60 .- 1 - 27 . 60.-2 -27.6O._14l 
+ 16.60.-27.60.-1-4.60.-2 -27.60.-19} 
- 27. 60 .+ 32 . 60.-1-11. 60.-2 -27. 60.-24} 
2880 L J 
-^1 - 4 . 60 .- 11 . 60.-1 + 0 .60.-2 - 27 .60.-29} 
2880 ; J 
- 11 . 60 .- 36 . 60.-1 + 5 . 60.-2 -27. 60.-34} 
-jo. 60.+ 5.60._i-20. 60.-2 -27.6O._39 
2880 
20 
' 2880 V 
20 f 
'2880 I ’ 
20 
2880 I 
+ 5 . 60 .+ 16 . 60 .- 1 - 27 . 60.-2 - 27 . 60.-44 
2880 L 
?Lj_ 20 . 60 .- 27 . 60.-1 + 32 . 60.-2 - 27 . 60.-49 
2880 I 
^-27.60.-4.60.-1-11.60.-2 -27. 60.-54} 
I +32.60.-11.60.-1-36.60.-2 - 27 . 60.-59} 
2880 l 
20 
’^0 I 
Assembling, finally, the several portions, X, Y, Z', Z", Z'", of which 11(07) consists, 
and reducing those periodic functions, which have 5 and 10 for their period, to a 
period of 60, we see 
3 H 2 
