INDETERMINATE ANALYSIS. 



Sympfis of IndeUrm'male Formvle. Syiwpfis of Indderminaie FormvU. 



Form. I. Equation ax — by = + c Form. 9. Equation ax' 4- bx + m' = s 



/General value of x = m i + , 



"^^ J P'-aq 



*/5 



I General value oi x = m b ± c q r (^^_^_ ^^j_ ^f ^ bq ^-zm pq 



I oiy = ma±cp \ ' p'-aq 



in wliicli expreflions m is indeterminate, and the values of J ^ _ wj>' -f am y* 



/> and y refult from the folution of the equation ap— L ^~ ^'- — j^ 



7 ~ ± '• »'here m, », and b are known ; and/i and j indcterminates. 



Form. 2. Equation ax +'ijf = ^- Form. 10. Equation jr= - Njr- = + I. 



f General value oi x =' c q - m b T Gen val of r = ^ /' + ?'^^>"'+ (^-yv^'N)" 



I y^ma~cp J " ' ^ 



Number of lohition« = — i ^- / J'= ■ nn — ' 



the quantities p and q being afcertained as above, alfo m where p and y arife from the equation /«' — N 9" =r + r, 



indeterminate. and m is indeterminate ; except that it mull be even or odd, 



as the cafe requires. See Prop. IX. 



Form. 3. Equation a x -^ by .^ c z = d Form. 1 1. Equation x* — N_)r^ = + A. 



f General value oi x = {d - ez) q - mb C Gen. val. of ;e = pm ± N y n 



X y=ma-{d~c^)p \ y^pn±qm, 

 the quantities p and q being found as above, alfo m inde- the values -of m and n being formed from the equation m' 



J — N n' = + A, and p and q from the equation ^' — N q- 



terminatc, and z any integer < - . = -f i . 



Form. 4. Equation .v' — ay- = »'. Form. 12. Equation ax' + i*-' -f- c x f /' = z'. 



Gen. val. of .V = / -r a 7- r„ . . . - c - ^bf 

 hich expreffions a is given, and /> and q are indeter- / Or, x = 1——^^ ^^ — i^ 



« = 2 * <7 I Particular val. of ^ 



/• - ''r 



{Pari 

 Or, 



inniated, that may be alTumed at pleafure. .. v4*y '^) 



where all the quantities a, b, c, andy are given. 



{ 



5. Equation .v' + ay- = %'. Form, 13. Equation a x' + i x' -{-ex'' + dx + f — a'. 



Gen. val. of x = f a f r _ (S ^/^ - 4 . ^/' + ^^) 8/^ 



>• = % Pi _ , I °' •" - 16. / - 64 a/^- 8^</-' + d^' 



where a, 3, c. Sec. are all known quantities. 



a being given, and^ and q indcterminates as above, 



Form. 6. Equations atc^ + bxy -f ji^ := z". Form. 14. Equation m" x^-]-bx^ + cr' + d .x + 



Gen.v:A.oix=ipq + bq f .v = l6r' ^'-64.^^-8 .*>'+ *' 



X — p"- + bpq + aq 



where p and y are ir.detcrminates, and a and b giv 



tities. 



where alfo m, b, c, &c. are all known quantities. 



Form."]. Equation a. V' 4- i.r = c'. Form. i^. Equation m'a-' -f i.r' + fx' + </.v +/'= a'. 



fGen. val. of. ^ -i?-- ( P-ticular val. of . ^ 'L^^"'^^J^ 



J p'-aq' J 4*/ +4'««/ 



1 */? I Or, 4mV j-_ 4m5/ • • 



l ^^F~-aT ^ ''-i^ + Sm'f-^m'c- 



, J ; 1 1 J ■ , In thefe exprefRons m, 3, c, &c. are -known quantities, and 



whore a and & are. known quantities, and * and n indeter- • 1 j . i u- r . n u ur j 



• ^ ^ ' r 1 with regard to the ambiguous fign, it mult be obferved, 



that when it is taken -f- in the numerator, it muli be — ia 

 the denominator, and the contrary. 



Form. 8. Equation m' .t" + i v + f rr 2=. Form. 16. Equation ax' -f b x'- + c a- 4- /^ = z'. 



Gen. val. of .. ^ . P -^1 ^ \ Particular val. of .v = ^ilzJ^JL?/!, 



bq — Zinpq «- 21 aj — C ■ 



^ _ "I p' 4 me g bpq' a, h, c, &c. being known quantities. 



b q' — 2 mpq'- 



whcre m, ^, and c are any given numbers, and/ and y indc- 

 terminates. 



Synopfii 



: I 



