THEORY OF POLYCONIC PROJECTIONS. Ill 



point at which the minimum is found lies, therefore, in the 

 Northern Hemisphere. 



The values of 2>m and p'm for a given value of n on any 

 given meridian would have to be determined by successive 

 approximations until the equation containing 2?m, p'm, X', 

 and n would be satisfied by the value obtained. For 

 particular meridians the ec^uation becomes much simpler. 

 Thus for the central meridian it becomes 



tan ^^ sin<Pm , 

 2 n 



When this value is substituted in the equation for the 

 second derivative, we obtain 



. , ri d'lci ri+cos<^„,-7in 



sm p^smp^ Lp 5pX = ^L ^2^sin-^. J' 



It is upon this meridian that we obtain the smallest of all 

 the minima. 



Let us now suppose n>l. The conditions are now 

 changed, since fc = at the poles. The value of Ic upon 

 each meridian passes through a maximum instead of a 

 minimum; this maximum is found in the Southern Hemi- 

 sphere and lies between the colatitude p^ and the South 



sm 7) olc 

 Pole. This is shown by the fact that — =r^ ^ is equal to 



n—1 for 2> = 0, a positive result; for p = Poj ^' = 9' and th-e 



value is —cos Pq, stiU positive, since Po>^', for p^ir the 



value becomes 1— n, a negative result. Hence the maxi- 

 mum lies between the straight line parallel and the South 

 Pole. 



When n is slightly greater than unity, it may happen 

 that, starting at zero, the value of Ic would pass through a 

 maximimi in the Northern Hemisphere; then it would fall 

 to a minimum in the same hemisphere, and finally pass 

 through a maximum in the Southern Hemisphere to return 

 to zero at the South Pole. This depends upon whether 



"COS 'd' 



^— ^ becomes greater than n; this may weU happen if 



cos Pjj^ 



7h is but slightly greater than unity. 



Lagrange proposed to profit by the fact that n and po 



were arbitrary parameters to so determine them that k 



would vary as slowly as possible at a given point upon the 



