913 



SUM, SUMMATION. 



SUN. 



911 



1-- 4 2~ 2 + 3- ; -f . . . , it will be convenient to begin from some term 

 which will make the series more convergent. Let a, = (10 + x)~ 3 : we 

 have then for 10-* + H~ : + . . . . the following : 



_L 1 _L 1 J_ li- I. L 



10 t2 10 2 * 6'lC 3 ~" 30 10* + 42 10' &C '' 





which may be easily calculated, and the preliminary series 1~' + . . . 

 + '.>- uiay then be added. 



l' may i/ueu m; iiuuet.i. 



7. The finite series a a + a^+a., + . . . + a*_i (x terms) is thus trans- 

 formed (D. C., p. 266) : 



/" 1 



J , * <fe - 2(*- 



~ SO [4] " + 42 



1 a,' a,', 

 6 2 



~ 30 [8] + 



ill which the detached coefficients are again the numbers of Bernoulli. 

 Or, if the sum of the series ad injinitum be known, s, the preceding 

 may be expressed as follows : 



/I 1 a.' 1 a.'" 



^ a,dx - - + 6 T - 30 TTT H 



But when the complete series is divergent, the set of terms 

 o u + a, -r . . . + a,_i may be thus expressed : 



f \ la.' 1 a,'" 



c + J^a.dx- 2 a -+ 6 "F ~ 30 "[*]" H &a ' 



wlitT.! c must be determined by an instance. Thus if we make 

 I + jr)-' we have for 1 + 2~' + . . . + x- 1 the following series : 



11 J_ 1 J_ 1 



c -rlg* (!-*)- 2 j^j-j - J2 (i + z )J + 120 (1 f *)' ~ 



Add U + l)-' to both sides, and write x-l for x, which gives for 

 1-' + . . 



1 



1 



T'> determine c, choose such a number for x as shall make this series 

 convergent, say r=10. Calculate !-'+... + 10->, term by term, 

 which i easily done, and equate the sum to 



1 

 1200000 ~~ ' 



1 1 



r + 2-302585093+ ~ 1200 + 



20 



which gives c- -5772156 . . . , the number mentioned at the head of 

 the table for series of inverse powers already given. With the value 

 of c thus determined, and a sufficient table of logarithms, the larger 

 tli,- iiiii.ib.-r of terms in 1~> + . . . +*-', the more easily is its approxi- 



ilue calculated. 



8. The series log 1 + log 2 + . . . + log x is of sufficient importance 



to have an article to itself. Make a, = log (1 + *), and proceed as in 



-t example, which will give for log (1 . 2 . 3 ... .<) the following 



c+logx.x-x+ 2 Io S* + 12l ~ 360? + - " 



c might be determined as before, but a particular mode of investigation 

 hows it to be log ( V2), where * = 3'14159 . . . , as usual. This gives 



a result of the greatest use, particularly in the more complicated 

 applications of the theory of probabilities. 



li. The series 0- + 1" + 2" + . . . +x*, in which a, = x* , n being 

 integer and positive, is by the case of 7, and adding x* to both sides, 



1^ nx"- 1 l_ n(w-l) (n 

 6 2 



z* 

 ~2 



__ 

 30 



2 .8.4 



but this vanishes when x = 0, whence c must be taken accordingly in 

 every instance. To take an example which shall require a little 

 extension of the series beyond the terms used above, let it be required 

 to find (P + V+ . . . +*'. Looking at the article NUMBERS OF BER- 

 NOULLI, we find a supply of coefficients in 



i -L L L 691 



6 30 42 30 66 2730' 

 and the ram required is 



8 + 2 + 6 2 30 



7.6.5** 



2.3.4 

 _!_ 7.6.5.4.3.2.1 

 30 2.3.4.5.6.7.8' 



7 .':.',. 4. 3*= 

 "2.3.4.5.6 



The Xapcrian logarithm, which ! always used in mathematical investi- 

 gations unless the contrary tin rxiirt-oBed. It is 2-302585093 x comm. log. 

 AHTS AKD SCI. DIV. VOL. VII. 



which vanishes when x= 0, whence 



\_ 7_.C_._5.4.3.2.1 _ 

 C+ 30 27374.5. 6. 7. 8 =0 ' 



and the rest may be reduced to 



a^ #* 7./' 7x* a? 

 8~ + 2" + TT~2T + 12' 



SUMACH, or SWAMP SUMACH (Rhus ToMcodcndron), a North 

 American shrub, possessing peculiar properties, from which it has 

 l>een named also the Trailing Poison Oak. The leaves, which are 

 trifoliate, thin, shining when fresh, of a dark-green colour, are the only 

 parts officinal in this country. But the leaves, branches, aud flowers 

 contain a milky juice which blackens on exposure to the air, aud may 

 be used as an indelible ink when applied to cotton or linen. Besides 

 this very acrid milk, the plant, when not exposed to the sun's rays, by 

 growing in the shade, or during the night, exhales a hydrocarburetted 

 gas, which acts very potently on persons of a peculiar susceptibility, 

 when exposed to it. In two or three days after touching or being 

 very near the plant, the skin inflames and swells, being attended with 

 intense burning pain. If the face be affected, the eye-lids are so tume- 

 fied as to close up the eyes ; and the whole head is swelled and covered 

 with little blisters containing serum. Occasionally the whole body is 

 enormously swollen and covered with similar vesicles. When the 

 inflammation and swelling have subsided, the skiu desquamates, and 

 an intolerable itching is felt for several days afterwards. These 

 symptoms closely resemble erysipelas, and are moderated by treatment 

 suited to that complaint. Professor Barlow states that the best appli- 

 cation is a weak aqueous solution of bichloride of mercury. All 

 persons however should avoid touching any sumach which has milky 

 juice. 



The leaves, or an extract of the inspissated juice of this plant, have 

 been recommended in several diseases, particularly herpes, paralysis, 

 and consumption. It is little used in this country, but it appears 

 entitled to some confidence in local paralysis, such as that of the jaw. 

 It must be given with caution, as large doses act like narcotico-acrid 

 poisons. 



SUMBUL, written also Sunbul, is a generic term applied to many 

 fragrant roots, one or more of which have been already noticed under 

 SPIKENARD. What is now to be noticed is a root recently introduced 

 into medical practice, of which there are two, perhaps three varieties, 

 the source of none of which is accurately known, though conjectured 

 to be the produce of some umbelliferous plant, to which the name of 

 A nyelica mosckata has been given. One is designated Russian sumbul, 

 another, Indian sumbul; while a third is called Cathayan sumbul, 

 which perhaps scarcely differs from the first kind, which is said to 

 reach Moscow through Kiachta, the same channel as the Russian 

 rhubarb. The Russian is the most powerful in odour, which is more 

 or less musky. The taste is bitter and slightly acrid. The analysis 

 shows it to contain a volatile oil, two balsamic resins, one soluble in 

 ether, the other in alcohol, with wax, starch, &c., and a peculiar acid 

 termed sumbulic. It yields its properties to alcohol or ether ; it is 

 also given in powder, watery infusion, decoction, or extract. The 

 balsamic resins and oil impart a stimulating power to it, which renders 

 it useful as an antispasmodic, and it is especially available in hysterical 

 affections ; and like other medicines of the same class, it would appear 

 to be sedative, as it calms nervous restlessness and procures sleep. It 

 has also been recommended in dropsy aud atrophy. 



SUMBULIC ACID. [ANGELIC ACID.] 



SUMMARY CONVICTION. [LAW, CRIMINAL.] 



SUMMARY JURISDICTION. [JUSTICES OF THE PEACE.] 



SUMMER. [WINTER, &c.] 



SUN (Latin, tul ; Greek, ijXios, Helios). In the present article we 

 confine ourselves to the astronomical characters of the sun's orbit, and 

 to what we tnow of its physical constitution. For more particulars on 

 the measurement of time, see TIME ; on chronology as dependent on 

 this body, and on the more common characters of its motion, see YEAR. 

 See also MOON ; SEASONS ; ASTRONOMY ; ZODIAC (on mythology) ; 

 ZODIACAL LIGHT ; TWILIGHT ; &c. 



It is needless to say that if the utility of the subject of an article 

 were to determine its length, the one we are now commencing ought 

 in justice to occupy several volumes of the work : were we, however, 

 seriously to mete out the importance of the sun in columns of a 

 Cyclopaedia, our panegyric would not be more quaint than that of Sir 

 John Hill, who says that this luminary is " enough to stamp a value on 

 the science to which the study of it belongs." In relation to astronomy 

 this is particularly true ; for it would be possible to preserve life on 

 the earth, and to keep order, without any knowledge of the moon, 

 planets, or stars ; but to do this without any acquaintance with the 

 sun's motions would be absolutely impossible. The source of light and 

 heat, and through them of the alternations of the vegetable world, is, 

 in the highest secondary sense, the giver and sustainer of life ; but this 

 very importance ensures names to so many results of solar phenomena, 

 that the present article is stripped of details, by their entering more 

 appropriately into others. 



