280 Mr. E. Howard Smart on the Third- Order 



Therefore by (15) and (16) putting, as in Whittaker's tract, 



Now the quantities f „ %, fci, X, and /* are the same 

 throughout the system: hence by summation 



W = ~2xt — a^ S0<+ a? ZUsUi 



+ x(x 2 + ^)Su < {e i w-i(i-^)}J . . . (22) 



Now correcting for the four errors above mentioned we 

 have in order 



2 0i=O, SCU*=0, 2©JJ; 2 =0,2-Y- - — \=0. 



A,r/ 

 But — p must be independent of X 2 , and /a 2 , since these 



depend on the square of the distance of the image-point 

 from the axis. 



is the condition for freedom from distortion for full pencils 

 to this order of approximation. The same result is obtained 

 by considering Ayi/i/i', This of course necessitates that 



