108 SUMMARY OF CURRENT RESEARCHES RELATING TO 



rality if it is developed in a series arranged according to the powers 

 of X. We, therefore, have 



fx= a + b \'" + cX" + e\P + etc., (3) 



in which a, h, c, etc., are constants, and the number of terms may be 

 taken as great as is desired. 

 Let us also put 



C = A, (a, - 1) + A^ (a^ - 1) + A3 (a, - 1) + etc. 

 D = Ai 61 + Aj 62 + A3 63 + etc. (4) 



E = Aj Ci + A2 Cj + A3 C3 + etc. 

 F = Ai ^1 + Aj ^2 + A3 63 + etc. 

 etc. etc. etc. 



the number of these equations, and the number of terms in the right- 

 hand member of each of them, being the same as the number of terms 

 in the right-hand member of (3). Now substituting for the yx's in 

 (1) their values in terms of the auxiliaries C, D, E, etc., of the equa- 

 tions (4), we find 



- = C + Da"' + Ea" + Fxp + etc. (5) 



Considering X as the abscissa, and / as the ordinate, this is the 

 equation of the focal curve. Its first derivative, with respect to / and 

 A, is 



^ = -/'(rnDx-'-i + nEX"-! + etc.), (6) 



which, as is well known, expresses for every point of the curve the 

 tangent of the angle made by the tangent line with the axis of 

 abscissas. The number of rays of different degrees of refrangibility 

 which can be brought to a common focus will evidently be the same 

 as the number of times that the focal curve intersects the focal plane. 

 But the focal plane is necessarily parallel to the axis of abscissas ; 

 and therefore the greatest possible number of intersections of the 

 curve with the plane can only exceed by one the niunber of tangents 

 which can be drawn parallel to the axis of abscissas. To find these 

 tangents we equate (6) to zero, and obtain 



= mD\"'-i + nE\"-i +etc. (7) 



As X can never be either zero, negative, or imaginary, we have 

 to consider only the real positive roots of this equation ; each of 

 which corresponds to a tangent. To make the number of tangents as 

 great as possible, the quantities D, E, F, etc., must be independent of 

 each other ; which will be the case when the right-hand members of 

 the equations (4) contain as many A's as there are powers of X in the 

 dispersion formula (4). All the terms of (7) contain the common 

 factor X"""^. Taking it out we have 



- mD = nEX"-" -h^jFxP-"' + etc., (8) 



from which it is evident that the number of real positive roots in (7) 

 will always be one less than the number of powers of X in (3), Hence 

 we conclude that : — 



