whence 



IX FOCAL PLANE OF TELESCOPE. 339 



9 



J<?(p)+Ji(p)= nearly. 





Vi 



~--'(-2a-o) \ A- 



■1-=^=^-^+*) 00 



arA" 



Putting z= ; , v y ^ , / l= ; , v ^ , *=« « 



and 



/= 2 /7i £ ^ «fr 



6 *r\2a + o) /„ \ 1 - Ay sm" J ?>/ V 1 — A'/ w»" y 



The calculation of these complete elliptic integrals can be effected 

 in the following manner. 



1-Vg^ /= 1 -Vjl 

 1 + V* ' 1_ Tc + l/P- 



» -4-+» (!)'+ »(!)•+ 



*-M*)'+ »(*)'+ 



with the relation log h . log h 1 =tt 3 , we have 



K= £ (l + 2/i + '2Ä 4 + 2Ä 9 + ) 



ü_ /,- A+ _ r l+2A+2Ä4+2Ä , + » 



2tt- A — 4A 4 +' à 9 - 



_ A' 1-2ä + 2/i 4 -2ä 9 + ' 



jf_ l+3 3 /t 1 - s + 5 2 A 3 - 8 + * 



4X L + ä u +ä"+ 



Either of these formulae enables us to calculate i? from K ; the 



intensity of illumination can therefore be easily calculated for both 



cases. 



* See Weierstrass-ScVrwarz, 'Formeln ztnd Lehrsätze zum Gebrauche der elliptischen Functionen 

 2te Auflage, Berlin, 1893— p. 44, p. 61. 



