The Project Gutenberg eBook #36640: Lectures on Elementary Mathematics

x^√2y − p −

q

2^√2y − p^.

Supposing then that the quantity y satisfies the equation

4(2y − p)(y² − r) = q²,

which developed becomes

py²
3


y − ₂ − ry +

pr q² _{= 0}

2 8

−

,

and which, as we see, is an equation of the third degree, the equation originally given may be reduced to the following by extracting the square root of its two members, viz.:

x² + y = x^√2y − p −

q

2^√2y − p^,

where we may take either the plus or the positive value for the radical ^√2y − p, and shall consequently have two equations of the second degree to which the given equation has been reduced

and the roots of which will give the four roots of the original

equation. All of which furnishes us with our first instance of the decomposition of equations into others of lower degree.

The method of Descartes which is commonly followed in the elements of algebra is based upon the same principle and consists in assuming at the outset that the proposed equation is produced by the multiplication of two equations of the second degree, as

x² − ux + s = 0 and x² + ux + t = 0,

where u, s, and t are indeterminate coefficients. Multiplying them together we have

x⁴ + (s + t − u²)x + (s − t)ux + st = 0,

comparison of which with the original equation gives

s + t − u² = p, (s − t)u = q and st = r.

The first two equations give

2s = p + u² + ^q , 2t = p + u² − ^q .
u

u

And if these values be substituted in the third equation of con- dition st = r, we shall have an equation of the sixth degree in u, which owing to its containing only even powers of u is resolvable by the rules for cubic equations. And if we substitute in this

equation 2y − p for u², we shall obtain in y the same reduced equation that we found above by the old method.

Having the value of u² we have also the values of s and t, and our equation of the fourth degree will be decomposed into two equations of the second degree which will give the four roots sought. This method, as well as the preceding, has been the occasion of some hesitancy as to which of the three roots of the reduced cubic equation in u² or y should be employed. The difficulty has been well resolved in Clairaut’s Algebra, where we are led to see directly that we always obtain the same four roots or values of x whatever root of the reduced equation we employ. But this generality is needless and prejudicial to the simplicity which is to be desired in the expression of the roots of the proposed equation, and we should prefer the formulæ which
you have learned in the principal course and in which the three roots of the reduced equation are contained in exactly the same manner.

The following is another method of reaching the same for- mulæ, less direct than that which has already been expounded to you, but which, on the other hand has the advantage of being analogous to the method of Cardan for equations of the third degree.

I take up again the equation