462 Mr. J. Prescott on. Connective Equilibrium 



values of m and a, the value of -j- "will appro;) eh some con- 



CLX 



stant as # becomes infinite ; i. e., the curve will have an 



asymptote. The only types of curve which will apply to 



the physical problem are those to which this asymptote 



is parallel to the axis of x and at a finite distance from it ; 



for the value of y when x is infinite is the value of p7~ l at 



the centre of the sphere. It is easier to start with x infinite 



clii 

 and - 7 zero and work towards the origin. I therefore began 



dx dv 



with y — 100, -r- =0 when x=ao , and proceeded by successive 



steps until y became zero. The values obtained are only 

 approximate ones, but I do not think the errors in the values 



of x and -- when y is zero are greater than 5 or 6 per cent. 



. ' dy 



The following table gives the calculated values of -— and y. 

 6 ^ dx d 





dy 







(hi 





X. 



dx 



y- 



100 



X. 



dx 



y- 



ex; 







21 



2-347 



72-07 



100 



•0333 



98-33 



20 



2-588 



69-60 



80 



•0525 



97-47 



19 



2856 



66-88 



60 



•0993 



95-95 



18 



3154 



63-87 



50 



•1689 



94-61 



17 



3-483 



6055 



40 



•3045 



92-25 



16 



3-844 



56-89 



36 



■5321 



90-78 



15 



4-232 



52-85 



32 



•8796 



88-28 



14 



4-649 



48-40 



30 



1-051 



86-38 



13 



5-085 



43-54 



29 



1-141 



85-29 



12 



5-529 



38-23 



28 



1-241 



8410 



11 



5960 



32-49 



27 



1-352 



82-81 



10 



6-352 



26-34 



26 



1-475 



81-41 



9 



6-670 



19-83 



25 



1-613 



79-87 



8 



6-877 



1306 



24 



1 : 767 



78-18 



7 



6-957 



6-14 



23 



1-939 



76-33 



6-10 



6-98 







22 



2-131 



7430 









The accompanying curve shows the relation between y and 

 x from the point where y = to ,y = 40. The numbers in the 

 table can easily be converted into distance from the centre 

 and density by means of the relations 



-IV 



cy 



4ttK( 7 -1) 



and 



P =f- 1 =r 



