2 The She'iltoth (questions) of Rab Alai (8th century) also belong probably to the school of Pumbeditha, though their author was not Gaon.
High plateaus like that of Pamir (the " Roof of the World ") and Armenia, and lofty mountain chains like the snow-clad Caucasus, the Alai, the Tian-shan, the Sayan Mountains, exist only on the outskirts of the empire.
The administration of kazas, or cantons, was usually entrusted to the cadis and the holders of the more important fiefs; the sanjaks, or departments, were ruled by alai beys or mir-i-livas (colonels or brigadiers), pashas with one horsetail; the vilayets, or provinces, by beylerbeys or mir-i-mirans (lord of lords), pashas with two horse-tails; these were all originally military officers, who, in addition to their administrative functions, were charged with the duty of mustering and commanding the feudal levies in war time.
The Alai range of the Pamir, continued by the Kokshaltau range and the Khan-tengri group of the Tian-shan, and the Sailughem range of the Altai, which is continued in the unnamed border-range of West Sayan (between the Bei-kem and the Us), belong to this category.
The border-ridges of the Alai Mountains, the Khantengri group, the Sailughem range and the West Sayan contain the highest peaks of their respective regions.
+(m -3) D 5(213) (214) (15) - (13) (14) (14), as= and and we see further that (alai +a2a2+...+amam) k vanishes identically unless (mod m).
ï¿½, and (alai +(72a2+a3a3+ï¿½ï¿½ ï¿½) P = (Pith +P2t2 +P3f 3 3+ ï¿½ ï¿½ ï¿½) P ï¿½ Instead of the above symbols we may use equivalent differential operators.
By similarly transforming the binary n ic form ay we find Ao = (aI A 1 +a2 A2) n = aAn A l = (alAi - I -a 2 A 2) n1 (a1ï¿½1 +a2m2) = aa a ï¿½ - A i n-1 A2, n-k k n-k k n-k k A = (al l+a2A2) (alï¿½1+a2ï¿½2) = a A ï¿½ =A 1 A2, so that the umbrae A1, A 2 are a A, a ï¿½ respectively.
= (A11+A22)n by the substitutions 51 = A l, E1+ï¿½1 2, 52 = A2E1+ï¿½2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = ï¿½la1 +ï¿½2a2ï¿½ We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c).
It was noted that Stroh considers Method of Stroh.-In the section on " Symmetric Function," (alai +a 2 a 2 +...
Now the symbolic expression of the seminvariant can be expanded by the binomial theorem so as to be exhibited as a sum of products of seminvariants, of lower degrees if alai 0-2a2 +...+crea0 can be broken up into any two portions (alai -1-0-2a2-1-ï¿½ï¿½ï¿½ +asas) +(as+1as +1 +o-8+2as+2+ï¿½ï¿½ï¿½ +ooae), such that Q1 +a2+...
The generating function is I - z2' 52 For 0 =3, (alai +a2a2+a3a3) 10; the condition is clearly a1a2a3 = A3 = 0, and since every seminvariant, of proper degree 3, is associated, as coefficient, with a product containing A3, all such are perpetuants.
The former is inconclusive, and in the latter the reading Ai/alai/cold is entirely conjectural; the name might equally well be Lysimachus or Lysias.
The Hissar range, a westward continuation of the Alai Mountains, separates the Zarafshan from the tributaries of the Oxus - the Surkhan, Kafirnihan and Vakhsh.
Rang Kul Lake occupies a central basin or depression; but the Kara Kul drains away north-eastwards through the Sarikol (as the latter, bending westwards, merges into the Trans-Alai) to Kashgar and the Turkestan plains.
Thence the province stretches northwards across the mountains of the Tian-shan system and southwards across the Alai and Trans-Alai Mts., which reach their highest point in Peak Kaufmann (23,000 ft.), in the latter range.
Of Khokand; the Isfairam (12,000 ft.), leading to the glen of the Surkhab, and the Kavuk (13,000 ft.), across the Alai Mts.
Trans-Alai, Alai, Kokshal-tau, Alexander range, Terskei Ala-tau, Kunghei Ala-tau, Trans-Ili Ala-tau and Dzungarian Ala-tau.
To the north-east, as far as the Alai valley - probably along the edge of the Pamir plateau.
The northern slopes of the Alai chain are richer in trees.
In the Kaufmann Peak, in the Trans-Alai range, and of 19,680 ft.
In the Alai range.
Kashgar has connexion with Ferghana and Bokhara over the Kyzyl-art pass (14,015 ft.) and down the Alai valley.
The southernmost range is the Trans-Alai, or rather its W.S.W.
Max Friedrichsen) assign both the Alai and the Trans-Alai Mountains to the Pamirs.
The other arm of the bifurcation, situated farther south, and beginning at the Terek-tau, is double; it consists of the Alai and Trans-Alai ranges, continued westwards in the Karateghin, Zarafshan, Hissar and Turkestan ranges, though orographically the Trans-Alai ought probably to be described as the border-ridge of the Pamir plateau.
At its south-western extremity the Kokshal-tau merges in the Kokiya Mountains (16,000-18,000 ft.), which at their other end are met by the Alai Mountains and the Terek-tau.
On the south the Ferghana valley is fenced in by the lofty range of the Alai, backed by the parallel range of the Trans-Alai.
The Alai is a well-defined ridge with steep slopes, and both it and the Terek-tau, which prolongs it towards the Kokshal-tau, are flanked next the Ferghana valley by what appear to be the old uplifted strata both of the old Palaeozoic series of metamorphic limestones and of the newer Tertiary series of softer conglomerates and sandstones.
The general altitude of both ranges is 16,000-19,000 ft., but the Trans-Alai culminates in peak Kaufmann (23,000 ft.).
The Trans-Alai is a true border range, the ascent to it from the Pamir plateau (13,000 ft.) on the south-east being gentle and relatively short, while both it and the Alai tower up steeply to a height of 11,000-14,000 ft.
The Alai valley is in ill repute because of the enormous masses of snow which fall in it in the winter.
The Alai Mountains are continued westwards in the radiating ranges of the Karateghin Mountains, Zarafshan Mountains, the Hissar Mountains and the Turkestan range, which reach altitudes of 18,500-22,000 ft., though peak Baba in the Zarafshan range reaches nearly 20,000 ft.
The Trans-Alai are continued in the Peter the Great range, which culminates in the Sandal group at close upon 25,000 ft.
In the Sary-jas, the Terskei Ala-tau, Khan-tengri, Alai, Trans-Alai, Terek range, Trans-Ili Ala-tau, Kunghei Ala-tau, Kokshal-tau, Dzungarian Ala-tau, Alexander Mountains and Talas-tau.
In the Alai region there are other extensive glaciers, e.g.
The shortest route, though not the easiest, between Kashgar and East Turkestan in the east and Ferghana and West Turkestan in the west is over the Terek pass or the pass at the head of the Alai valley, a dangerous route in winter by reason of the vast quantity of snow which usually accumulates there.