_a_ _c_ _ad_ + _bc_ (113) --- + --- = ----------- _b_ _d_ _bd_ _a_ _c_ _ad_ - _bc_ --- - --- = ----------- _b_ _d_ _bd_ _a_ _c_ _ac_ (118) --- × --- = ---- _b_ _d_ _bd_ _a_ _c_ _a_/_b_ _ad_ (121) --- divᵈ. by --- or --------- = ---- _b_ _d_ _c_/_d_ _bc_ 125. These results are true even when the letters themselves represent fractions. For example, take the fraction _a_/_b_ -------, _c_/_d_ whose numerator and denominator are fractional, and multiply its numerator and denominator by the fraction _e_ _ae_/_bf_ ---, which gives ----------, _f_ _ce_/_df_ _aedf_ which (121) is -------, _bfce_ which, dividing the numerator and denominator by _ef_ (108), is _ad_ ----. _bc_ But the original fraction itself is _ad_ _a_/_b_ (_a_/_b_) × (_e_/_f_) ----; hence ------- = --------------------- _bc_ _c_/_d_ (_c_/_d_) × (_e_/_f_) which corresponds to the second formula[17] in (124). In a similar manner it may be shewn, that the other formulæ of the same article are true when the letters there used either represent fractions, or are removed and fractions introduced in their place. All formulæ established throughout this work are equally true when fractions are substituted for whole numbers. For example (54), (_m_ + _n_)_a_ = _ma_ + _na_. Let _m_, _n_, and _a_ be respectively the fractions _p_ _r_ _b_ ---, ---, and ---. _q_ _s_ _c_ Then _m_ + _n_ is _p_ _r_ _ps_ + _qr_ --- + ---, or ----------- _q_ _s_ _qs_ and (_m_ + _n_)_a_ is _ps_ + _qr_ _b_ (_ps_ + _qr_)_b_ ----------- × ---, or ---------------- _qs_ _c_ _qsc_ _psb_ + _qrb_ or -------------. _qsc_ _psb_ _qrb_ _pb_ _rb_ But this (112) is ----- + -----, which is ---- + ----, _qsc_ _qsc_ _qc_ _sc_ _psb_ _pb_ _qrb_ _rb_ since ----- = ----, and ----- = ---- (103). _qsc_ _qc_ _qsc_ _sc_ _pb_ _p_ _b_ _rb_ _r_ _b_ But ---- = --- × ---, and ---- = --- × ---. _qc_ _q_ _c_ _sc_ _s_ _c_ Therefore (_m_ + _n_)_a_, or (_p_ _r_ )_b_ _p_ _b_ _r_ _b_ (--- + --- )--- = --- × --- + --- × ---. (_q_ _s_ )_c_ _q_ _c_ _s_ _c_ In a similar manner the same may be proved of any other formula. [17] A formula is a name given to any algebraical expression which is commonly used. The following examples may be useful: