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The form of the three given equations shows that a, p, y are the three roots 
of the equation 

* "+T^+^T=1. 

a+s b+s c+s 

in which s is regarded as the unknown. On clearing of fractions, and arranging 
in the form of a cubic equation in e, it is seen that the sum of the three roots is 
-(a + 6+c) + (x+y+z). 

Hence a+.p + y= — (a+b+ c) -\-(x-\-y+z) , and x-\-y + z— a-\-a-\-b-\-^-\-c-\-y. 

Note. It may be of interest to state that if each letter be squared the re- 
sult expresses the distance of any point from the origin in terms of ellipsoidal 
curvilinear coordinates. 

1S6. Proposed by B. F. FIHKEL, A.M., M.Sc, Professor of Mathematics and Physics, Drury College, Spring- 
field. Mo. 

(z-\-x)a— (z— x)b=2yz....(l); (x-\-y)b—(x—y)c=2zz....(2); (y + z)c— 
(jy—z)a=2xy....(Z'). Find the values of x, y, and z by the method of linear si- 
multaneous equations. 

Solution by (J. B. M. ZEEE, A.M., Ph. D., Professor of Chemistry and Physics, The Temple College, Philadel- 
phia, Pa. 

Let x=^(b-\-c)u, y—i(a+C)v, z—$(a-\-~b)w. 
.-. (a— b)w-\-(b+c)u=(a + c)vw....(l). 
(b— c)w+(a+c)«=(a+&)MW....(2). 

(c— a)v+(a + b)w=(b+ c)uv....(3). 

We might eliminate v, w and get an equation of the fifth degree in u. We 
will, however, proceed as follows: Add (1), (2), (3), then 

aw(2— u— «)+&m(2— v— w)-\-cv(2— u— w)=0. 

This is the case when u=v=w=0; or u=v—w=l; or m=0, w=v=2; or 
d=0, m=w=2; w=0, u=v=2. 

The first two sets of values satisfy the conditions. 
.-. x=y=z=0; x=i(b+c), 2/=K a +c), z=i(a+b). 

Notb. This is exercise 31, page 224, Systems of Linear Simultaneous Equations, of Fisher 
and Schwatt's Higher Algebra , and has given teachers of algebra throughout the country considerable 
trouble. Solving the equations for a, 6, and c, we readily find that 

b=x—y+z, and 
e=x+y— a. 
.-. x=h(b+c), y=i(a+c), z=J(a+6), as one set of values for a;, y, and z. Editor F. 
Also solved by L. C. WALKER .