In using the rule to convert _a_ + _j_ _b_ to R∠θ, the index (45°) of the B scale is set to the larger component and the cursor to the smaller component, on scale A. Then θ (or its complement if _b_ is greater than _a_) is read on B under the cursor. The cursor is then set to θ on the C scale, and R is read on A under the cursor. The rule is made by Messrs. John Davis & Son, Limited, Derby. THE SOLUTION OF ALGEBRAIC EQUATIONS. The slide rule finds an interesting application in the solution of equations of the second and third degree; and although the process is essentially one of trial and error, it may often serve as an efficient substitute for the more laborious algebraic methods, particularly when the conditions of the problem or the operator’s knowledge of the theory of equations enables some idea to be obtained as to the character of the result sought. The principle may be thus briefly explained:—If 1 on C is set to _x_ on D (Fig. 38), we find _x_(_x_) = _x_^2 on D under _x_ on C. If, however, with the slide set as before, instead of reading under _x_, we read under _x_ + _m_ on C, the result on D will now be _x_(_x_ + _m_) = _x_^2 + _mx_ = _q_. Hence to solve the equation _x_^2 + _mx_ − _q_ = 0, we reverse the above process, and setting the cursor to _q_ on D, we move the slide until the number on C under the cursor, and that on D under 1 on C, _differ by m_. It is obvious from the setting that the _product_ of these numbers = _q_, and as their difference = _m_, they are seen to be the roots of the equation as required. For the equation _x_^2 − _mx_ + _q_ = 0, we require _m_ to equal the _sum_ of the roots. Hence, setting the cursor as before to _q_ on D, we move the slide until the number on C under the cursor, and that on D under 1 on C, are _together equal to_ _m_, these numbers being the roots sought. The alternative equations _x_^2 − _mx_ − _q_ = 0, and _x_^2 + _mx_ + _q_ = 0 are deducible from the others by changing the signs of the roots, and need not be further considered. [Illustration: FIG. 38.] EX.—Find the roots of _x_^2 − 8_x_ + 9 = 0. Set the cursor to 9 on D, and move the slide to the right until when 6·64 is found under the cursor, 1·355 on D is under 1 on C. These numbers are the roots required. The upper scales can of course be used; indeed, in general they are to be preferred. EX.—Find the roots of _x_^2 + 12·8_x_ + 39·4 = 0. Set the cursor to 39·4 on A, and move the slide to the right until we read 7·65 on B under the cursor, and 5·15 on A over 1 on B. The roots are therefore −7·65 and −5.15.