600 SCIENCE PROGRESS 



Table III. Newtonian Coefficients. 



If x* + ibx* + yx + d = y = y 3 , then— 



x = y - b + {b* - c}/y - {2b 3 - $bc + </}/3y 8 + \zb h - 5^ + £»a* + 3^ - ^}/3y 4 - 



- \7lfi - 2i£V + i$b*c* + 4(JW- 3^ - bbcd + d i }\3 i y b - etc - 



Table IV. Centred Equations. 



1. If x l + cr + d = y — y 3 , then — 



x= y- c\ 3 y~ dlsy - cd\$y k - {$d-i*}\f-f - {3 1 .2cd* - c*} / tfy 7 - $ {d* - <*d} 3*y* - 



- 7 {3» . 2^ 3 - c*d}/3 e y 10 - 2 {3 8 . 5^ 4 - 30A/ 8 + 4^} /3V 11 - etc. 



2. If x* + ex* + dx -{- e =y = y 4 , then — 



* = y - c/4y - djtf- {fe - 2c 3 } I tfy 3 - 2 {${d 2 + 2ce) - c 3 }U i y i -2{4de - A/}/4 3 y 6 - 



-2{4 s .3r , -4 s .3(^ + ^)+5^}/4V-2{4 s .5(^V + ^)-6o^^ + 7^}/4V- 



- {4 8 . bde* - tf{cd 3 + 3c 2 de) + 6^}/4 s y 10 - 7{4 4 • 2^ - 4 2 . 2(3<V + ^ 4 ) + 

 + 4o(2<rV 2 + c*e) - i8f 6 } /4V - etc. 



3. If x* + ex 3 + dx 1 + ex +f — y = y 5 , then — 



* - y - f/5.V - dlsy' - {5' - ' ! }/sV - {5/- ^}/5V - We + cf) - 2c 3 d) / 5 y- 



- {5V + 2#) - i&Pc + c»tf) + 2^}/5V - {s» . 3^- S (d 3 + bede + 3c 3 /) + 

 + 7^}/5V- {5* • 2/ J - 5 3 • 2(^ + ce 9 + zedf) + ly'd 3 - \\c')\tfy* - etc. 



Table V. 

 If ^ = y n and ^ = £/»y, C = c\ny\ D = d/n^ 3 , . . ., then— 

 y-i* = y -i,/,-y = 1 - {B + C + D . . .} + {(n- i)B* + (n- 3)0 + (n- s)D* . . .+ 



+ («-2)2^C+(«-4)CZ> . . ,}/2! -{(»-2)(2«-2)i5 3 + («-5)(2«-5)C 3 . . .+ 



+ (n- 3 )(2« - 3)3^ ! C ...+(»- 7)(2« - 7)3CZ> 2 ...+(»- 5X2* - 5) 

 6^^C. . .}/3 ! + {(» - 3 )(2» - 3 )(3« - 3)^ 4 • • • + (»-4)(2» - 4)(3» - 4) 

 4B 3 C. . .}/4! -etc. 



rata P7. 

 If/r = ^K-r-y" = O ; and H^fy\ny n , and A"= log(^y/y ?i ), then— 



1. x - y - y/f-f y[£> y (yo) + no]N'j2\ - y[Z> y (yO) + «o][£> v (yO) + 2«o]ZT 3 /3! + etc. 



= y-y/* r +y{ 2 y/'y//y + (i -n)}H*\2\- y{3yV"y/fy + Wvlfy) 3 + * 

 + 9(i - nftf'ylfy +(" - ")(i - 2tt)}/P/3\ + etc. 



2 . ^ = y _ ^y/^'y . ^+ fy/^'y . Dyi^jfy) . A' 2 / 2 ! - V^ /*' y • [^(V'y/^'y)] 3 - K*l3 1 + etc. 



= y - *y/*'y ■ ^ + ^y/^'y . { i - Vt/^V • VyWv}*'/* ! - 

 - *y/*'y { i - >h/^'y • Vylfyf + (Vt/^V • Wy)' - (Vn'/^'y) 2 • (fWy)} 



A^ 3 /3 ! + etc. (if y^'yjn^y > o and < 2). 



