192 Proceedings of the Royal Society of Edinburgh. [Sess. 
For purposes of approximate quadrature this may be evaluated in 
the form 
^ 27 ra 2 ej sin (0 - /3)d(6 — p) — ^ 27ra 2 e{cos (0 — p)^ - cos (0 - p) 9 } 
where the positions (1) and (2) indicate the beginning and end of a narrow 
zone on the spherical surface at P, and where e is the average value of e over 
this zone. The average values over the zones bounded by the successive 
values of 0 — /3 were obtained graphically from the curve showing the 
relation between 0 — /3 and the energy. They are given in the sixth 
column of Table VIII. 
The final calculation in shown below. 
0-p. 
Cos (6 — /3). 
Difference. 
Average 
Energy. 
Product. 
6 
4 3 
9 9 
14 1 
19 19 
25 19 
32 36 
42 38 
60 25 
1 
•9976 
•9872 
•9702 
•9437 
•9040 
•8425 
•7238 
•4000 
•0024 
•0104 
•0170 
•0265 
•0397 
•0615 
•1187 
•3238 
•98 
•927 
•835 
•730 
•570 
•390 
•240 
•070 
•0024 
•0096 
•0142 
0193 
•0226 
•0240 
•0285 
•0227 
Sum -1433 
This fraction multiplied by ra 2 represents the energy passing into 
the nucleus expressed in terms of the energy per unit surface incident on 
the nucleus along the ray EC. 
The whole energy supplied in terms of the same quantity is 
2t r(b - af( 1 - cos p), p = 23° 35'. 
Hence the ratio of the condensational wave energy which passes into the 
nucleus to the amount which falls on it is 
27 ra 2 x*1433 16 x ’1433 
27 r(b - a) 2 x ( 1 - cos P) 36 x *0835 
The condensational energy passing through the nucleus will emerge 
again into the shell ; but in this case the angles of incidence will be 
included between the values 0° and 51° 50', which is the angle of total 
reflection for the condensational wave (see Table VII and diagram III of 
fig. 8). The proportional loss of energy at this second refraction will be 
