TRANSACTIONS OP THE SECTIONS. 19 



If we consider any number of quantities, say four, -whose product xyzu is equal 

 to a contstant jij, their sum, S, = ,r+»/+s+M, is the smallest possible when they are 

 all equal ; for if two of the quantities are unequal, we can diminish their sum Si 

 by replacing each by the square root of their product. From these well-known 

 principles we can deduce the solution of a celebrated question, known as Huyghens's 

 problem, as follows : — 



2. The sums of the products of these quantities, taken two and two and three 

 and three, viz. 



S.;^=xy^xz+m-\-yz-^yu-\-zu, S^ = xyz+xyu+xzu-]-t/zu, 



are composed of terms which, multiplied together, give a constant product jj,. S3 

 and S, will therefore have their smallest possible values when the terms are equal, 

 viz. when x=y=z = u. 



3. The product, 



V=a + x)il+i/)Q.+z)(l+u), which =l+Si+S,+S3+i>, 



is the least possible when x=y=z=u, because then Sj, Sj, S3 have their minima 

 values. 



4. The expression 



ftXYZ 



^ - («+X)(X+Y)(Y-hZ)(Z+6)' 



in which a and b are constants and X, Y, Z variables, can be written 



and the greatest value of H, or the least value of the denominator of the expression 

 last written, coiTesponds to 



X_Y_Z_6 

 a - X ~ Y ~ Z' 

 for the product 



X Y Z J ^ 6 



a "X ' Y ■ Z a 



is constant. 



5. The preceding argument can evidently be extended to any number of vari- 

 ables. We are led to seek the maximum of a quantity analogous to H and con- 

 taining n variables in treating the well-known question (Huyghens's) : — " Let there 

 be any number of perfectly elastic balls ranged in a straight line ; the first strikes 

 the second with a given velocity, the second with the velocity communicated by 

 the first strikes the third, the third strikes the fourth, and so on. The masses a 

 and h of the first and the last being given, determine the masses of the intermediate 

 balls that the last may receive the maximum velocity." M. Picart has treated this 

 question by means of the difierential calculus (' Nouvelles Annales de MathtSma- 

 tiques,' 1874, pp. 212-219) ; but the investigation is long. The mathematicians 

 who had solved it previously (Huygheus, Lagrange, &c.) have, he states, only de- 

 monstrated that H was really a maximum in the case of three balls. The present 

 note contains a simple and complete solution by means only of elementary algebra. 



On the sinr/ular Solutions of Differential Equations of the First Order ivhich 

 represent Lines at Infinity. By Paul Mansion, Professor in the University 

 of Ghent. 



1. The following is a resinnS of the theory of singular solutions of differential 

 equations of the first order. 

 (I.) If a difierential equation, 



fi^,y,y')=0, or cf,(x, y, x')=0, (1) 



