ON LINES OF EQUAL INTENSITY. 9 



Neglecting powers of £, r h higher than the 4th, and putting in 

 the values of the differential coefficients found above in the expression 

 for à I, we find 



/ap\ s /aj*\% 4 , /a? 3 i<p a<*\ a^ a?* £3 

 oI== VW VW s + iJ Va a - a*, ô y ô «/ a* »z * * 



Ka ^ a a ^ a y a ^> a ^ a a ~i fcl ., 

 TF iy + ~äy TF>) + aa ~ây~â~z~ yrJ ç " r ' 



/a? a^ a^ a^>\ a^ a^ /*yV/j? W 



+ " Va,r 2>2/ + a y ax/ a? »y* T V»y/Wv/' 



~ La x a * ? + V a * a ^ + a y a * / ? ' ^~ a y a # ' J 



Thus, the equation to the locus of equal intensity ol becomes 



T^TT S + U~Fay + TyTyM + ây a y T-±Vài t 



which represents two central conies. 



Referred to principal axes, the equation to the curve is trans- 

 formed into 



where 



(A-B)Ç'* + (A+B)f* = ± J\dl =±G. 



a ^ a a ^ a 

 a x a # a?/ a?/ 



I / a^> ô j^ a ç? a y / a ^> a ây a ^ \ 2 



~ V \y^ tj + ~ay "ay/ + \T7 iy " *Y ~**) ' 



