August 1, 1889.J 



KNOWLEDGE 



213 



(7) If P=any perfect number and p any prime number 

 (except 5), then (P'~;y'')/5 is always an integer. 



Many other interesting deductions may be drawn. 



Secondary Perfects are as difficult to obtain as the primary. 

 The following are all I have been able to obtain easily : 

 P., = 2\3.7=672. Pa = 29..3.11.31 = .523776, Pj = 

 2'''.3. 11. 43.127., P2=28.5.7.19.37.73. This means that the 

 sum of the factors of 672 equals twice the number itself, thus 

 e.g. the sum of 



1,2,3,4,6,7,8,12,U,16,21,24,28,32,42,48,.56,84,96,112,168, 

 224,336^2x672. Some Perfects the sum of whose factors 

 equals three times the Perfect itself are : P, = 2''.3''..5.7 ; P3=: 

 2'*.3.5.7.19.37.73; P3=2'.3«.5.17.23.137.547.1093. 



III. A pretty theorem of Sylvester's is as follows. 

 Every number may be resolved into a sequence in as many 

 ways as it contains odd factors ; e.;/., 63 has 6 odd factors, 

 viz., 1, 3, 7, 9, 21, 63, and therefore it may be represented 

 by the six following sequences and no others : — 63=:31 +32 

 =20 + 21 +22=S + 9 + 10+ 11 + 12 + 13 = 6 + 7 + 8 + 9 + 111 

 + ll + 12 = 3 + 4 + .5 + 6 + 7 + S-(-9 + 10 + ll. On similar 

 principles we are able to deduce that n'" is the sum of ii 

 consecutive odd numbers; e.g., 4'':=13 + 15 + 17 + 19. Or 

 again, n™ equals a sequence of re natural numbers if n be 

 odd and m greater than unity. Thus, e.g., 



.52=3 + 4 + .5 + 6 + 7. 



IV. We may form squares by means of cubes in various 

 ways. One well-known equation is Si=(2J,)'". This 

 means that if we take any number of successive cubes from 

 unity their sum is always a square ; e.g., 13 + 23 + 3''' + 4''' + .5^ 

 = (l + 2 + 3 + 4 + 5)'^=15'^. 



Another equation is 2''' + (2S')'=(3S'")'" ; e-g-, m=4 gives 

 us P + 23 + 33 + 43+{2(l + 2 + 3 + 4)}3 



= {3(12+02 + 32 + 40)12^902, 



Again, P + 33 + 53 + {in-\f = {nM)- where n is the 



denominator and M the numerator of an}' odd convergent of 

 v/2; eg.,n=5 M=7, then 1-' + .3H-'5-' + 7'' + 9''=(l + 3 +5 

 + 7 + 9)(72)=35-^. 



While we are dealing with series, the following equations 

 will, I think, be found interesting :—S., + S-=2S32. Thus 

 n = 3 gives us 1 '^ + 2-' + S'* + 1 ^ + li^ + 3^ = 2 {( T' + 2-' + 3^ )} 2. 

 Other equally pretty equations are: — 7.S|; + 5Si:=12S,,S3 

 (3S, + 2S;)5/5S.,=S;j/S2. &c. Thus if we suppose n=2, we 

 have 



7(l« + 26) + 5(r< + 2')=12(12+22)x(I3 + 23) 



i.e., 7x65+5x17 =12 x5 x9, or 540=540. 



Again, for the other equation let w=3, and we get 

 3( l--. + 2'' + 3>) + 2(1 + 2 + 3--')^5(lH 2' + 3')={V' + 2^ + 3-') 

 + (l^ + 22 + .3-^) 



i.e., 3+276 + 2x216 —5 + 98 =26-=-14 



or 12(iO H-490 =36--14, 



which is true, and so on for all v.alucs of n. 



V. It is impossible to find any two cubes, whose sum or 

 dilference is a cube, or even twice a cube, but there arc 

 various ways of resolving a cube into three cubes. Euler's 

 well-known formula is somewhat elaborate. t)no of the 

 simplest is — 



{b{b^ + 2a3)y=[a{h-'-ay,'+ ;/-(//' -«■')) ■'+ {a{2b-' + a-')]\ 

 where a and b may be any inte^^ers whatever. 



To get the sum of three cubes into three other cubes, a 

 simple formula is (3^)- — (y2^:i_j_^^,2_|_^2_|_,(_^,i^)3_j.^^j2_j_,^2_ 

 'lp'iy=(p'^ + ?-)•' + {3p^ + :i</f + ( p- - 3</-') '^a formula which 

 may be generalised — Here p and 1/ may be any integers. 



Lot;)=5, ^=3, then (3 x52-.32V' + (5'^ + 32 + 4x5x3)3 

 + (52 4. 32 _ 4 ^ 5 X 3)--'= (52 + 1]^ + 3(52 + T) + 52 - 3 X 3-^)''. 

 Thus, after dividing by 2, we get 



l-l + 33:. + 47:i=|;j:!+i7:. + 5p^ 



and similarly for all values of /; and y. 



y^e can resolve a cube into a number of squares. 



Let X:=p^—'ipr/'^, y=^p'q—q^. 



Then 



,-^ + y^=(p^- + ff= fp(p2 + q2y.2+ {lip' + q'r-- 

 and generally 



(p2 + ,/2 + ,2 + .,0J3={p(p2 + ^2 .+ ,2)12 



+ Mp'+'?' — 2')}'+ {'•(?'+'?'—+-')! '+*C. 



Thus, if jo = 3, q = '2, r=\, we get 

 (32 1-22+ 12)3= {3(32 + 2»+12)}2 ^ [2(.32 + 22+l')}2 



+ {1(3^ + 22+ 1 2)} 2, or 14'' = 422 + 2S2 + 142. 



We can resolve a number into three cubes. Let n = the 

 number-. Assume 7i^={n—b.ry^ + {c.r+\Y + (rLK~\Y, and 

 make b={c + d)jn'^. 



Then x= [^(d'-c^-nb')l{,P + c^-h^)\; 



e.f/. ?l=:2, 6=:3, C=:7. '!='>. 



Then a;=-f-, and 203=73 + 143 + 17 '. 



If 6=c, then 33 + 43 + 53=53. 



If b=d,then 183 + 193 + 213 = 283; 



c=d, gives us 93 + 103 = 13 + 123, &c. itc. 



THE FACE OF THE SKY FOR AUGUST. 



By Herbert Sadler, F.R.A.S. 

 POTS on the .solar surface .are bt^coming 

 lather more frequent, but there have not 

 been many of any consideraUe size as j-et. 

 ( 'onveniently observable minima of Algol 

 occur on the 4th at lOh. 39m. p.m., and on 

 the 27th at 9h. 10m. p.m. The variable 

 star o (Mira) Ceti will arrive at a maximum 

 on the 6th. Mercury comes into superior conjunction with 

 the sun on the 7th, and is practically invisible throughout 

 the month. Venus is a morning star, rising on the 1st at 

 111. A.M., with a northern declination of 20° 51 ', and an 

 apparent diameter of 18 V'. On the Last d,ay of the month 

 she rises at Ih. 30m. a.m., her declination having decreased 

 to 19^°, and her diameter to 15". She passes from Orion 

 through Gemini into Cancer. At about 4h. a.m. on the 

 18th she will be 26' due north of the 4th magnitude star 

 ^ Geminorum. Mars is .actually a morning star, as he 

 rises on the 1st at 3h. 6m. a.m., and on the 31st at 

 2h. 58m. a.ji., but his diameter does not exceed 4^" dui-ing 

 the month, so that he is pr.actically not worth looking at 

 with a telescope. His path throughout the month is in 

 Cancer. Saturn is invisible, being in conjunction with the 

 sun on the 16th. Jupiter is visible during all the working 

 hours of the night, but owing to his great southern declina- 

 tion is very unfavourably suited for observation. He rises 

 on the 1st at 5h. 20m. p.m. with a southern declination of 

 23° 22', and an apparent diameter of 44i", and on the 31st 

 at 3h. 17m. p.m., with an apparent diameter of 41 '. He is 

 nearly stationary throughout the month in Sagittarius, and 

 during the last fortnight will bo found about 23' nearly 

 due north of the 5i magnitude star 4 Sagittarii. On the 

 evenings of the 1st and 2nd a 9th magnitude star will be 

 seen a little to the north of the planet and its satellites. To 

 his occultation by the moon on the 7th we allude below. The 

 following phenomena of the satellites occur between the times 

 of the planets' being S° above the horizon, and the sun's being 

 8° below, on the days naiiiod. On tlie tth an egre.ss from 

 transit of the third satellite at lOh. 49m. p.m., and the 

 ingress of its shadow at 11 h. Him. On the 6th a transit 

 ingi-ess of the first satellite at 8h. 59m., and ingress of the 

 shadow of the satellite at 9h. 56m., the satellite itself 

 passing oft' the planet's disc at llh. 16m. t)n the 7th a 

 reappearance from eclipse of the first satellite at 9h. 30m. 33s. 

 On the 10th an occultation (disappearance) of the second 

 satellite at lOh. 3m. On the 12th an egress of the shadow 

 of the second satellite at 9h. 52m. On the 13th a tran.sit 

 ingress of the first satellite at lOh. 48m. On the 15th an 



