1x3y 43 3yee 3ayy 41+42+43|44|y3+a3+3eexy+a=y+a’ II. 7+8+9+10|11 a+e+y+uxe=a+yxe+y 2015 aayy=e+ Theo. XXIII. Theo. XXIV. Theo. XXV. 2 Theo. XXVI. 20 aaee: ee+yy=eeyyyy+uu 21 ee ee 1x2222ey=2au 2 25€+y=e—y2+4au or +4ey (Step 1.) Theo. XXVII. 26aaeaae 2xe 27 eee yae 4xu 29 uue yyu 26+27+28+29 30 aa +ee+yy+uuxe=ae÷yuxa+y But 2 a2+e2+y2+u2 : ae+yu=a+y: e Theo. XXVIII. |32|a-|-e-|-y-|-u : e+y=aty:e (Step. 12.) ··33a2 + e2 + y2+u2: a‡e+y+u=aeyu:ety Th. XXIX. Step. 1434a2+e2 + y2+u2 : ce+yy=ae+yu: au or ey Theo. XXX. 2 2 2 Then 563aee+aey=a2×3y-\-u Then 583aeeeeea1×3y+u But 60 ay2: e2=u2 : y2 2 : 2 Theo. XXXVI. Theo. XXXVII. ··61e2: a2=y2: e2=u2: y2=3)--u: 3ate Theo. XXXVIII. | 62 |a3 +-3a2è=a3 ++-3a2e=aa×â-\-3ê 586263 a3+3a2e+3ae2+e3—a2×3y|-u+a+3e 64+65+66+67 68 eyau or eyxatu+3×e+y 69a3aaa 44 70e3aau 45 71yauu 3 69+70+71+72 73 @3 +e3+y1‡u3=a+uxaˆ+u2 44x4 aaau acee 45x75 auuu-uyyy Theo. XXXIX. Theo. XL. Theo. XLI 4xat 76 O=aeyy-aeeu 2XU 77 =uyee uyya` T 74475+76+77: 74+75+76-7778 aux aa+uu—ae3-+-aey2+uye2‡uy3—aue2—auy2 Whence 79a-u: e2+y=ae+uy-au orey: au or ey 2 Theo. (XLII. 13+14+15+16 17 exy+b+yxa+b=e2+uxa+y+u 11xy 18 byy yuu (Theo. XXXIV.) Theo. XLVI Theo. XLVIL Theo. XEVI. Theo. XLIX. 25, 27.28bbaa: uu-ee=a+b: y Theo. L. ya=ee |29|ya=ee 29+30+3132yxa+2y+b=u+e* 3, |34|b2+a2=b2+a2 Theo. LI. 34+35+36 37 62+ua+y2+e2+a2=bxb+y+a+axya Theo. LII. |39|u3byubbe 40 y3 euy bee (In. 190, 223, 224.) 38+39 &c. 4363+u3+y3+e3+a3=bxbb+be+ee+aaxu‡a 34, 35 |44|b*+a+=b++a+ 45\u*+e+=y2b2+y•a2 3646 ya2b2 Theo. (LIII. 37• 47b++u*+y*+e*+a*=b*xb* +y*+a* +a*xy2+aa (Theo. LIV. After the fame Manner may innumerable other Theorems be raised, which the Learner may purfue at his own Difcretion, whofe Ufe will be seen in the Effection of the following Problems. PROBLEM LXV. 623. To find three Quantities in, a, e, y, whereof the Sum of the Extream's a+y=s, and the Sum of the Cubes of the faid Extreams a3+y3 |