THE THEORY OF FISHER. 187 



A similar situation occurs with respect to the effect of accretion on the 

 moment of inertia and period of rotation of the planet. The moment of 

 inertia is 





l=-i^n I pr*dr 



or, measured for convenience in terms of a unit equal to -^^^ G. G. S. 

 units, ^ /-j9 



p^sin/Jd^ 







sini9 / 



which reduces to 



7=2/33— |- (6- i?^) (46) 



Now, in the same units the stratum itself adds to the moment of inertia 

 the quantity 



<57=/3M/?=i3^|l+(^y|d/3 (47) 



But this is in excess of the true increment, because the differential conden- 

 sation diminishes the moment of inertia of the underlying mass. The true 

 increment by (46) is 



which may be written: 



dl =8I-2^'K (^- 1) d/3 (48) 



where 



giving for comparison: 



K <" 



P'B 



dl ^^ 2K/SK 



(^-l)(l-Z)) (49) 



dl /? V ^ 



which is tabulated in column 7 of table 3. Then by combination of equa- 

 tions (45) to (49) may be computed 



dlog7^£ _±_ dl . . 



dlog/3 I ' 1-D ' dl ^ ^ 



which gives the ratio of the percentage increase of the moment of inertia to 

 the percentage increase of the radius. This would indicate also the percent- 

 age change in the length of the day if the stratum were deposited entirely 

 under normal incidence, or if the moments of momentum of the planetes- 

 imals with respect to the existing axis of rotation exactly compensated 

 each other. If, however, it be supposed that variations in the rotation, or 

 even the existence of the rotation, were brought about by lack of such com- 

 pensation, then the equations allow this effect to be distinguished from that 

 due merely to changes in the moment of inertia. 



