412 The Astronomer Royal on a Projection for Maps 



product of the evil by the extent of surface which it affects, 

 omitting the general multiplier <£, is 



{(i-ox^- 1 )'}*^^ 



Consequently the summation of the partial evils for the whole 

 map is represented by 



Or if r—6=y, and if we putj? for ~i, the expression is 



and this integral, through the surface to which the map applies, 

 is to be minimum. 



7. This is a case of the Calculus of Variations. The function 

 V to be integrated exhibits valuer for the differential coefficients 



M -^ V N- r ^ P-^. 

 U -d0> dy> V ~dp 



The equation of solution is N W =0. Now in consequence 



of the existence of a value for M, we cannot adopt the facilities 

 of solution which present themselves when M = 0, and we must 



therefore take the equation N— --U- =0 without modification. 



Here 



and the equation N— ~~ becomes 



y + d^m\6 . a dhj Q dy _ 



r-g sm 0._^_ cos 0.-^ — 0; 



sintf d9 2 dQ ' 



or 



sin 2 O.-^+sinO. cos0.-^ —y = 6— sin 0. 



8. For 0— sin# put the more general symbol ©, To solve 

 the equation, assume z— sin . ~ + y. Then, by actual differ- 

 entiation and substitution, 



